^  A^^r^   y^r^  ^w^  Chicago,  ^ 

1870. 


=^<2>*= 


Ivisofiy  Blakenian^   Taylor  dr»  Co.^s  Publications. 

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ROBINSON'S  MATHEMATICAL  SERIES. 

-+* ■- : 

FIRST  LESSONS 


m 


MENTAL  AND   WRITTEN 


ARITHMETIC 


ON  THE  OBJECTIVE  METHOD. 


EDITED   BY 

SAMUEL  D.  BAEE,  A.M. 


NEW   YOEK: 

IVISON,   BLAKEMAN,  TAYLOR,   &  COMPANY, 

138  &  140  Grand  Street. 

CHICAGO:  133  &  135  STATE  STREET. 

1871.    . 


^5- 5-6" 


Entered  according  to  Act  of  Confess,  in  the  year  1870,  by 

D.    W.   FISH,    A.  M., 
In  the  office  of  the  Librarian  of  Congress,  at  Washington. 


EDUCATION  DEPT. 


Smith  &  McDougal,  Electrotypers,  82  &  84  Beekman  Street,  New  York. 


PEEF  ACE. 


The  author  calls  attention  to  the  following  points  as  among  the 
claims  made  for  this  book. 

1.  It  treats  Number  objectively y  and  by  a  method  believed  to  be 
new  and  simple. 

2.  The  principles  and  processes  are  unfolded  in  natural  order^ 
as  occasion  demands  them. 

3.  The  greatest  pains  have  been  taken  in  the  attempt  to  present 
clearly  the  most  elementary  and  fundamental  principles,  in  the 
firm  belief  that  no  sound  scholarship  can  be  erected  on  any  other 
basis,  and  that  the  elements  are  not  only  the  most  important  but 
are  also  the  most  difficult  to  teach  successfully.  If  a  pupil  ever 
needs  guidance  and  help  it  is  when  he  sets  out  in  the  path  of 
knowledge.  Therefore,  the  aim  has  been  so  to  instruct  and  en- 
courage him  in  his  first  efforts  that,  gaining  strength  at  each  step, 
he  shall  advance  with  increasing  interest  and  delight. 

4.  Special  attention  has  been  devoted  to  Notation  and  Numera- 
tion. Numbers  consisting  of  three  periods  have  been  developed 
objectively.  It  is  believed  that  children  can  use  numbers  having 
nine  figures  as  readily  and  intelligently  as  those  having  three 
figures,  if  Notation  and  Numeration  be  understood.  A  careful 
examination  of  the  method  used  is  earnestly  solicited. 

5.  Throughout  the  book  the  pupil  is  taught  to  use  his  reason 
and  common-sense,  and,  as  a  rule,  is  not  required  to  memorize 
until  he  perceives  the  truth  of  what  is  to  be  committed  to 
memory. 

6.  From  the  commencement  mental  and  written  work  are  com- 
bined. At  the  outset  the  pupil  is  set  to  working  on  the  black- 
board. He  first  obtains  his  results  mentally,  and  afterwards 
reproduces  his  work  with  his  hands,  and  looks  upon  it  with  his 
eyes.  It  then  becomes  to  him  a  reality.  What  is  traced  by  the 
muscles  and  pictured  on  the  eye  is  the  more  indelibly  imprinted 


f7i24952;a 


IV  PREFACE, 

on  the  memory.  It  is  for  this  reason  that  the  book  contains  so 
much  work  in  the  form  of  Equations,  with  the  work  so  varied 
and  so  full.  It  is  believed  that  the  pupil  will  thoroughly  master 
the  Tables,  learning  readily  and  accurately  to  perform  all  the 
elementary  operations,  by  this  process  sooner  than  by  any  other ; 
since  he  must  use  the  Tables  at  every  step. 

7.  Addition  and  Subtraction  are  treated  in  immediate  connec- 
tion ;  also  Multiplication  and  Division.  Thus  their  correlations 
are  more  clearly  shown.  Multiplication  is  at  first  worked  up 
under  Graded  Addition,  and  Division  under  Graded  Subtraction, 
till  they  are  well  understood,  and  Tables  have  been  made  and 
used,  before  the  terms  Multiplication  and  Division  are  given.  In 
Division  the  three  terms  are  written  in  precisely  the  same  order 
as  the  corresponding  terms  in  Multiplication,  that  Long  Multipli- 
cation may  illuminate  and  illustrate  Long  Division. 

8.  The  Tables  for  Addition,  Subtraction,  Multiplication  and 
Division  are  developed  objectively  and  progressively. 

9.  The  principles  of  Factoring  have  been  so  applied  as  to  sim- 
plify Division  and  pave  the  way  to  Fractions. 

10.  The  rule  for  Addition  and  Subtraction  of  Fractions  having 
unlike  Denominators  is  developed  objectively. 

11.  Names,  Definitions  and  Rules  are  given  with  the  full  con- 
viction that  the  best  time  to  give  them  is  when  we  give  the 
things  named  or  defined,  and  the  method  of  operation  desoibed  in 
a  Mule.  If  a  child  cannot  comprehend  a  method  of  operation,  he 
should  not  be  required  to  use  it ;  but  if  he  can,  then  he  can  com- 
prehend a  clea.r  statement  of  the  method  in  a  Rule  ;  and  since  he 
understands  the  Rule^  and  it  is  important,  he  should  learn 
THE  Rule. 

The  aim  has  been  not  to  load  down  the  pupil  with  Arithmetic 
as  a  burden  from  without,  but  to  cause  it  to  spring  up  within  him 
by  a  natural  and  healthful  process  ;  that,  growing  and  unfolding 
with  his  intellect,  it  may  be  an  organized,  vital  and  indestructible 
part  of  himself. 

The  author  has  written  this  book  on  his  own  plan,  using  his 
own  methods  and  illustrations,  striving  to  adapt  them  to  the 
comprehension  of  children.  He  is  not,  however,  so  vain  as  to 
presume  that  he  has  made  no  mistakes,  and  has  produced  a  work 
above  criticism. 

S.  D.  B. 

June  1, 1870. 


SUGGESTIONS    TO    TEACHERS. 


The  author  of  this  book  requests  most  earnestly  that  teachers  will  seek 
fully  to  comprehend  his  plan  and  methods,  and  strive  to  work  by  them,  teach- 
ing the  book  precisely  as  it  is  written. 

It  is  not  taken  for  granted  that  the  pupil  knows  anything  about  Number. 
The  attempt  is  made  to  teach  each  thing  in  its  proper  place,  accompanied  by 
such  reasons  and  explanations  that  the  pupil  shall  compreMnd  tJie  sui^ect^  and 
not  be  required  merely  to  memorize  words  and  Tables  to  him  almost  without 
meaning.  The  work  is  begun  at  the  very  basis,  in  the  belief  that  the  first  and 
most  important  work  is  to  lay  broad  foundation-stones,  so  firmly  that,  whoever 
shall  build  upon  them,  he  shall  build  in  confidence,  knowing  that  he  is  on  the 
solid  rock. 

The  aim  is  so  to  instruct  the  pupil  at  the  commencement  that  he  shall  be 
able  to  use  the  knowledge  gained.  The  first  few  Lessons  are  quite  simple, 
since  he  is  to  be  put  to  working  on  the  blackboard  at  once.  When  each  figure 
and  sign  is  first  given  he  should  be  drilled  in  making  it  on'the  blackboard  till 
he  can  write  it  neatly  and  with  facility. 

The  Signs  +,  — ,  =,  and  the  Equation  by  Addition  and  that  by  Subtraction, 
are  carefully  developed,  with  the  full  meaning  and  power  of  each  item,  so  that 
the  pupil  shall  at  once  be  able  to  use  them  understandingly.  By  their  use  he 
will  be  able  to  write  on  the  blackboard  from  day  to  day  all  he  learns  of  Arith- 
metic, and  to  reproduce  it  in  every  possible  form.  Pupils  love  blackboard- 
work  ;  and  in  no  other  manner  can  they  be  taught  so  successfully. 

Particular  attention  should  be  given  to  the  relation  between  Addition  and 
Subtraction  as  shown  in  Lessons  IX,  X,  XI,  XII  and  XLIX. 

The  method  of  developing  the  Tables  for  Addition  and  Subtraction,  and 
those  for  Multiplication  and  Division,  by  the  use  of  cubes,  and  the  manner  of 
reading  the  Tables  from  the  cubes,  should  be  carefully  explained. 


VI  SUGGESTIONS  TO   TEACHERS, 

The  method  of  adding  columns  of  numbers  by  6's,  shown  on  page  27,  really 
embraces  the  substance  of  Addition.  The  pupil  should  be  well  drilled  also  in 
adding  by  7's,  8's,  9's  and  lO's.  This  will  lay  a  good  foundation  for  subsequent 
work  in  Notation. 

The  greatest  possible  pains  should  be  taken  in  the  development  of  Notation 
and  Numeration,  as  set  forth  on  pages  46,  47,  62,  68,  74-76. 

Addition  by  objects,  as  shown  on  pages  53  and  53,  and  Subtraction  by 
objects,  as  given  on  pages  56  and  57,  should  be  explained  till  every  pupil  can 
point  out  every  step. 

The  Lessons  on  Graded  Addition  and  Subtraction,  commencing  on  page  81, 
are  specially  hnportant,  since  they  lay  the  foundation  for  Multiplication  and 
Division,  and  enable  the  pupil  to  see  that  what  foUows  is  but  a  new  applica- 
tion of  what  he  has  already  learned. 

For  each  of  the  numbers  2  and  3  two  Multiplication  Tables  are  given ;  but 
for  each  of  the  numbers  4,  5,  6,  7,  8,  9,  10,  only  one  Multiplication  Table 
is  given.  In  each  case  a  second  Table  should  be  written  out  by  each  piipil,  in 
the  manner  explained  in  Lesson  LXIV,  and  be  learned  with  the  one  given. 

The  Examples  and  Explanations  showing  the  precise  manner  in  which 
Multiplication  is  derived  from  Addition,  and  Division  from  Subtraction, 
should  be  dwelt  upon  till  every  pupil  can  solve  the  same  Example  both  by 
Addition  and  Multiplication,  or  by  Subtraction  and  Division. 

Great  pains  should  be  taken  to  show  clearly  the  relations  between  Multipli- 
cation and  Division  by  the  Solutions  and  Explanations  given  in  the  book. 
The  corresponding  terms  in  Multiplication  and  Division  are  written  in  the 
same  order ^  that  the  pupil  may  the  more  readily  understand  the  relations. 

The  difference  between  the  two  cases  in  Division  given  on  pages  96  and  100 
should  be  carefully  pointed  out. 

The  subject  of  Factoring  as  applied  to  Division,  especially  for  the  purpose 
of  deducing  the  General  Principles  of  Division,  should  receive  special  atten- 
tion, since  these  Principles  are  important  in  Fractions. 

It  is  left  to  the  Teacher  to  give  all  needed  Explanations  in  Denominate 
Numbers,  and  to  supply  any  further  Examples  needed. 

THE  AUTHOR. 


rwTW' '  o'^s^'^^s^ 


LESSON   /. 

1.  In  this  picture,  how  many  girls  are  in  the  swing  ? 

2.  How  many  girls  are  pulling  the  swing  ? 

3.  If  you  count  both  girls  together,  how  many  are 
they? 

One  girl  and  one  other  girl  are  how  many  ? 
Jf.  How  many  kittens  do  you  see  on  the  stump  ? 

5,  How  many  on  the  ground  ? 

6,  How  many  kittens  are  in  the  picture  ? 

One  kitten  and  one  other  kitten  are  how  many  ? 

7,  If  you  should  ask  me  how  many  girls  are  in  the 
swing,  or  how  many  kittens  are  on  the  stump,  I  could 
answer  aloud,  Oiie;  or  I  could  write  One;  orthus,  1. 

8,  If  I  write  One,  this  is  called  the  word  One, 

9,  This,  1^  is  named  a  figure  One^  because  it 
means  the  same  as  the  word  Oiie,  and  stands  for  One. 


8  FIRST  LESSONS  IN 

10,  Write  1.    What  is  this  named  ?     Why  ? 

11,  A  figure  one  may  stand  for  one  girl,  one  kitten,  or 
one  any  thing. 

12,  When  children  first  attend  school,  what  do  they 
begin  to  learn  ?    Ans,  Letters  and  words. 

15,  Could  you  read  or  write  before  you  had  learned 
either  letters  or  words  ? 

H-,  If  we  have  all  the  letters  together,  they  are  named 
the  Alphabet. 

16,  If  we  write  or  speak  words,  they  are  named  Lan- 
guage. 

16,  You  are  commencing  to  study  Arithmetic ;  and 
you  can  read  and  write  in  Arithmetic  only  as  you  learn 
the  Alphabet  and  Language  of  Arithmetic.  But  little 
time  will  be  required  for  this  purpose. 


LESSON   II. 

1,  If  we  speak  or  write  words,  what  do  we  name  them, 
when  taken  together  ? 

2,  What  are  you  commencing  to  study  ?  Ans,  Arith- 
metic. 

3,  What  Language  must  you  now  learn  ? 
If.,  What  do  we  name  this,  1  ?     Why  ? 

5,  This  figure,  1,  is  part  of  the  Language  of  Arith- 
metic. 

^.  If  I  should  write  something  to  stand  for  Two — 
two  girls,  two  kittens,  or  two  things  of  any  kind,  what 
do  you  think  we  would  name  it  ? 

7.  A  figure  Two  is  written  thus :  2.  Make  di  figure 
two, 

8.  Why  do  we  name  this  2i  figure  tivo  9 

9.  This  figure  two  (2)  is  part  of  the  Language  of 
Arithmetic. 


3IENTAL  AND    WRITTEN  ARITHMETIC, 


10.  In  this  picture  one  boy  is  sitting,  playing  a  flage- 
olet. What  is  the  other  boy  doing  ?  K  the  boy  stand- 
ing should  sit  down  by  the  other,  how  many  boys 
would  be  sitting  together?  One  boy  and  one  other 
boy  are  how  many  boys  ? 

11.  You  see  a  flageolet  and  a  yiolin.  They  are  mu- 
sical instruments.  One  musical  instrument  and  one 
other  musical  instrument  are  how  many  ? 

12.  I  will  write  thus:  112.  We  say  that  1  boy 
and  1  other  boy,  counted  together,  are  2  boys ;  or  are 
equal  to  2  boys.  We  will  now  write  something  to  show 
that  the  first  1  and  the  other  1  are  to  be  counted 
together. 

13.  We  name  a  line  drawn  thus,  — ,  a  horizontal 
line.     Draw  such  a  line.    Name  it. 

H.  A  line  drawn  thus,  |  ,  we  name  a  twrtical 
line.    Draw  such  a  line.    Name  it. 


10 


FIRST  LESSONS  IN 


Plus  means  more; 


15.  Now  I  will  put  two  such  lines 
together;  thus,  +.  What  kind  of 
a  line  do  we  name  the  first  ( — )  ? 
And  what  do  we  name  the  last  (  |  )  ? 
Are  these  lines  long  or  short  ? 
Where  do  they  cross  each  other  ? 

IS.  Each  of  you  write  thus:  — , 

11.  This,  +,  is  named  JPlus. 
and  +  also  means  more. 

18.  I  will  write, 

One  and  One  More  Equal  Two. 

19.  Now  T  will  write  part  of  this  in  the  Language  of 
Arithmetic.  I  write  the  first  One  thus,  1;  then  the 
other  One  thus,  1.  Afterward  I  write,  for  the  word 
More,  thus,  +,  placing  the  +  between  1  and  1,  so  that 
the  whole  stands  thus :  1  +  1.  As  I  write,  I  say,  One 
and  One  more. 

W.  Each  of  yon  write  1  +  1.  Eead  what  yon  have 
written. 

21.  This  + ,  when  written  between  the  I's,  shows  that 
they  are  to  be  put  together,  or  counted  together,  so  as 
to  make  2. 

22.  Because  -f  shows  what  is  to  be  done,  it  is  called 
a  Sign.  If  we  take  its  name.  Plus,  and  the  word  Sign, 
and  put  both  words  together,  we  have  Sign  Plus,  or 
Phis  Sign.  In  speaking  of  this  we  may  call  it  Sign 
Plus,  or  Plus  Sign,  or  Plus. 

28.  1,  2,  + ,  are  part  of  the  Language  of  Arithmetia 

Write  the  following  in  the  Zanguage  of  Arithmetic  : 

2Jf.  One  and  one  more.  25.  One  and  two  more. 

26.  Two  and  one  more. 


MENTAL  AND    WRITTEN  ARITHMETIC. 


11 


LESSON      IIL 
OJV-B  AJV^    OJVJ^  MO^Ii,    BQZTAL     27fO. 

1,  I  will  write  the  above  thus :  1  +  1  equal  2. 

2,  In  length,  are  these  lines,  =,  equal  or  unequal  ? 

3,  We  will  use  two  lines  thus  drawn,  =,  to  mean 
equal,  in  place  of  the  word  equal.  Writing  them  in 
1  +  1  equal  2,  we  have  1  -{-  1  =  2. 

Ji-,  Because  these  two  lines  thus 
written  show  something,  they  are 
called  a  Sign.  And  because  they 
mean  equal,  we  name  them  the 
Sign  of  Equality. 

5.  Anything  written  that  means 
something,  as,  for  instance,  1  +  1 
=  2,  is  called  an  Expression. 

6,  An  expression  like  the  above,  in  which  we  use  the 
Sign  of  Equality,  is  named  an  Equation* 


12 


FIE  ST  IjKJSSO.XS    IiY 


LESSON   IV. 

1,  How  many  boys  are  in  this  boat  ? 

^.  If  the  boy  in  the  end  of  the  boat  should  fall  out, 
how  many  boys  would  be  left  in  •  the  boat  ?  One  boy 
from  two  boys  leaves  how  many  boys  ? 

3,  Instead  of  saying,  One  hoy  from  two  hoys,  we  will 
say,  Tivo  hoys  less  one  hoy,  which  means  the  same. 
Then,  two  boys  less  one  boy  will  equal  how  many  ? 

^.  We  will  write,  2  less  1  =  1.     Eead  this. 

5,  We  wish  something  that  means  less,  to  write  be- 
tween 2  and  1,  for  the  word  less. 

6.  Aline  written  thus,  —  ,is  used 
to  mean  less,  instead  of  the  word 


7.  The   name   of   this  line,    — , 
when  thus  used,  is  iif^mie^.  Minus        ^f^i^^^y^c^^ 
means  less.  ^^^^^/^ 

8,  We  may  write  2  less  1  =  1,  or  2  —  1  =  1. 


MENTAL   AAJ}    WRITTEN  ARITHMETIC,  13 


iQXJS^'flQ^-^- 


9.  Since  this,  — ,  shows  that  something  is  to  be  done, 
what  may  we  call  it  ?    Ans.  A  Sign. 

What  is  its  name  ? 

10.  If  we  put  the  two  words  Sign  and  Minus  together, 
they  make  Sign  Minus,  or  Minus  Sign,  In  speaking 
of  this  Sign  we  may  call  it  Sign  Minus,  or  Minus  Sign, 
or  Minus, 

11.  Each  of  you  write,  2  —  1  =  1.  Eead  aloud  what 
you  have  written. 

12.  What  is  the  name  of  this  Expression :  2  —  1  =  1? 

13.  Write,  and  name  in  order,  the  following :  1,  2,  +, 
=,  — .     Of  what  do  these  form  part  ? 

Write,  in  Arithmetical  J^anguaae,  the  fottowinff : 

H.  One  equals  one. 

15.  One  and  one  more  equal  two. 

16,  Two  less  one  eaual  one. 


14 


FIRST  LESSONS  IN 


LESSON    V. 

1,  In  this  picture  how  many  boys  do  you  see  fishing? 
How  many  hunting  ?  How  many  in  all  ?  2  boys  and 
1  other  boy  are  how  many  ? 

2.  We  make  a  figure  Three  thus :  3. 

S.  1  boy  and  2  more  are  how  many  boys  ?    Are  1  boy 
and  2  more  just  as  many  as  2  boys  and  1  more  ? 
Jf,  How  many  birds  are  on  this  tree  ? 

5.  If  1  of  them  should  fly  away,  how  many  would  be 
left  ?     1  bird  from  3  birds  leaves  how  many  birds  ? 

6.  If,  instead,  2  of  the  birds  should  fly  away,  how 
many  would  be  left  ?     3  birds  less  2  birds  are  how  many  ? 

7.  1  from  3  leaves  how  many  ?     2  from  3  ? 

8.  3  boys  are  how  many  more  than  2  boys  ?   Than  1  ? 

9.  Write,  and  read  aloud,  the  following  Equations: 
1  +  1  =  2;  2-1  =  1;  2+1  =  3;. 

1  +  2  =  3;         3-1  =  2;         3-2  =  1. 


MENTAL  AND    WBITTEN  ARITHMETIC. 


15 


LESSON    I/A 

1,  In  this  picture  you  see  1  driver?  How  many  other 
men  in  the  wagon?  1  man  and  3  other  men  are  how 
many?     3  men  and  1  other  man  are  how  many  ? 

2,  We  write  a  figure  Four  thus:  4. 

3,  Are  3  men  and  1  man  more  just  as  many  as  1 
man  and  3  men  more  ? 

Jf.  If  the  driver  should  jump  from  the  wagon,  how 
many  men  would  be  left  in  the  wagon  ?  1  man  from  4 
men  leaves  how  many  men  ? 

5.  If,  instead,  the  3  other  men  should  jump  out,  how 
many  would  be  left  in  the  wagon  ?  3  men  from  4  men 
leave  how  many  men  ? 

6.  4  men  are  how  many  more  than  3  ?    Than  1  man  ? 

7.  How  many  horses  are  in  1  span  ? 

8.  How  many  spans  of  horses  are  drawing  this  wagon  ? 

9.  How  many  horses  are  there  in  all  ? 


16  FIRST  LESSONS  IN 

10.  2  horses  and  2  other  horses  are  how  many  horses  ? 

11.  If  the  2  horses  in  front  should  be  unhitched  and 
driven  away,  how  many  would  be  left  ?  2  horses  from  4 
horses  leave  how  many  horses  ? 

12.  4  horses  are  how  many  more  than  2  horses? 
How  many  more  than  3  horses  ?     Than  1  horse  ? 

IS.  3  horses  and  1  horse  are  how  many?     1  horse  ' 
and  3  horses  ?     2  horses  and  2  other  horses  ? 

14^.  1  horse  from  4  horses  leaves  how  many  ?  3  horses 
from  4  leave  how  many  ?     2  from  4  ? 

15.  2  liorses  are  how  many  less  than  4  horses  ? 

16.  1  horse  is  how  many  less  than  4  horses  ? 

17.  What  Language  have  we  commenced  learning  ? 

18.  Write  these:  1,  3,  2,  4,  +,  —,  =.    Name  each. 

19.  Of  what  Language  are  they  part  ? 

Write,  in  ;Arith77tetical  Za^iguage,  the  following : 

20.  One  and  one  more  equal  two  (1  +  1=2); 

21.  Two  less  one  equal  one ;    . 

22.  Two  and  one  more  equal  three ; 
2S.  One  and  two  more  equal  three ; 
2 If..  Three  less  one  equal  two ; 

25.  Three  less  two  equal  one  ; 

26,  Three  and  one  more  equal  four ; 
21,  One  and  three  more  equal  four ; 

28.  Two  and  two  more  equal  four ; 

29,  Four  less  one  equal  three ; 

50,  Four  less  three  equal  one  ; 

51,  Four  less  two  equal  two  ; 

52,  The  Sign  +  shows  that  what  is  written  at  the 
right  of.it  is  to  be  coimted  with  what  is  written  before  it. 

SS.  The  Sign  —  shows  that  what  is  written  at  the 
right  of  it  is  to  be  tahen  away  from  what  is  written  be- 
fore it. 


MENTAL  AND    WRITTEN  ARITHMETIC,  17 


LESSON    VII. 

i.  In  this  picture,  Minnie  has  1  rose  in  her  left  hand ; 
how  many  has  she  in  her  right  hand  ? 

2.  If  she  should  put  them  all  in  her  right  hand,  how 
many  would  she  have  in  her  right  hand  ? 

8,  We  make  a  figure  Five  thus :  5*    Make  one. 

^.  4  roses  and  1  rose  more  are  how  many  roses?  1 
rose  and  4  more  roses  are  how  many? 

5.  Are  4  roses  and  1  more  just  as  many  as  1  and  4 
more? 

6.  Willie  has  3  roses  in  his  right  hand;  how  many 
has  he  in  his  left  hand  ?     If  he  should  put  them  all  in 
his  right  hand,  how  many  would  he  then  have  in  his . 
right  hand?     3  roses  and  2  roses  are  how  many?     2 
roses  and  3  roses  are  how  many? 

7.  Are  3  roses  and  2  more  just  as  many  as  2  and  3 
more? 


18  FIRST  LESSONS  IN 

8,  If  Minnie  should  give  her  teacher  the  rose  in  her 
left  hand,  how  many  would  she  have  left  ?  1  rose  from 
5  roses  leaves  how  many  roses  ? 

9,  If,  instead,  she  should  give  away  the  4  roses  in  her 
right  hand,  how  many  would  she  have  left  ?  4  roses 
from  5  roses  leave  how  many  ? 

10,  If  Willie  should  give  his  mother  the  2  roses  in 
his  left  hand,  how  many  would  he  have  left  ?  2  roses 
from  5  roses  leave  how  many  ? 

11,  If,  instead,  he  should  give  her  the  3  roses  in  his 
right  hand,  how  many  would  he  have  left?  3  roses 
from  5  roses  leave  how  many  ? 

i^.  How  many  more  roses  has  Willie  in  his  right 
hand  than  in  his  left?     3  are  how  many  more  than  2  ? 

13,  How  many  more  has  Willie  in  his  right  hand 
than  Minnie  in  her  left  ?  3  are  how  many  more 
than  1  ? 

14'  How  many  more  roses  are  in  Minnie's  right  hand 
than  in  Willie's  ?     4  are  how  many  more  than  3  ? 

15,  How  many  more  has  she  in  her  right  hand  than 
Willie  in  his  left?     4  are  how  many  more  than  2? 

16,  How  many  more  has  she  in  her  right  hand  than 
in  her  left  ?     4  are  how  many  more  than  1  ? 

17,  How  many  roses  are  on  the  rose-hush  ? 

18,  How  many  would  be  left  if  Minnie  should  pick  1  ? 
If  she  should  pick  4?     If  2?     If  3? 

19,  5  are  how  many  more  than  4  ?  Than  1  ?  Than 
3?     Than  2? 

Wtite,  in  A.rlthnietical  language ,  the  foUowing  : 

4  and  1  more  equal  5 ;  5  less  1  equal  4 ; 

3  and  2  more  equal  5 ;  5  less  2  equal  3 ; 

2  and  3  more  equal  5 ;  5  less  3  equal  2 ; 

1  and  4  more  equal  5 ;  5  less  4  equal  1. 


MENTAL  AND    WRITTEN  AKITHMETIC. 


19 


LESSON    VIII. 

1.  How  many  chickens  are  on  this  hen's  back? 

2,  How  many  other  chickens  are  about  her  ? 
S,  5  chickens  and  1  chicken  are  how  many? 

4.  We  make  a  figure  Six  thus:  6.    Make  one. 

5.  How  many  ducklings  are  swimming  in  this  stream? 

6.  How  many  are  on  shore  ? 

7.  How  many  ducklings  in  all?  4  ducklings  and  2 
more  are  how  many?     2  and  4  more  are  how  many? 

<^.  How  many  swallows  are  on  this  gate  ? 

9.  How  many  others  are  on  the  fence  ? 

10,  3  swallows  and  3  other  swallows  are  how  many  ? 

11.  If  a  hawk  should  fly  away  with  the  chicken  on 
the  hen's  back,  how  many  chickens  would  be  left? 
1  chicken  from  6  chickens  leaves  how  many  ? 

12,  If  a  fox  should  steal  the  2  ducklings  on  shore, 
how  many  ducklings  would  be  left? 


20  FIRST  LESSONS  IN 

13.  If,  instead,  the  4  ducklings  should  float  away, 
how  many  would  be  left  ?  2  ducklings  from  6  duck- 
lings leave  how  many  ?     4  from  6  leave  how  many  ? 

14'  If  the  3  swallows  on  the  gate  should  fly  away, 
how  many  would  be  left?  3  swallows  from  6  leave 
how  many  ? 

JVrlte  t?ie  following  hi  £Jquations  : 


5  and  1  more  equal  6 
4  and  2  more  equal  6 
3  arid  3  more  equal  6 
2  and  4  more  equal  6 
1  and  5  more  equal  6 


6  less  1  equal  5 
6  less  2  equal  4 
6  less  3  equal  3 
6  less  4  equal  2 
6  less  5  equal  1. 


15.  Eead  aloud,  in  concert,  what  you  have  written. 

16.  James  had  3  pennies,  and  his  father  gave  him  2 
more  ;  how  many  had  he  then  ?  He  found  1  more ;  how 
many  had  he  in  all  ? 

11.  Flora  had  4  nice  dresses  for  her  doll,  and  her 
mother  made  2  new  ones  for  it;  how  many  dresses  for 
her  doll  had  she  then  ? 

18.  Charlie  caught  3  trout,  and  Willie  3 ;  how  many 
did  both  catch  ? 

19.  Mary  had  2  canaries  that  sang,  and  4  that  were 
not  singers ;  how  many  canaries  had  she  in  all  ? 

20.  Albert  had  6  pennies,  and  gave  3  of  them  for  6 
r.pples ;  how  many  pennies  had  he  left  ?  He  ate  2  of 
the  apples ;  how  many  had  he  left  ?  His  sister  Helen 
ate  2  more  of  them ;  how  many,  in  all,  did  Albert  and 
Helen  eat  ?     How  many  apples  had  Albert  then  left  ? 

21.  4  and  1  are  how  many?  2  and  3?  4  and  2? 
3  and  3  ?     1  and  5  ?     2  and  4?  '  3  and  2  ?     5  and  1  ? 

22.  2  from  5  leave  how  many  ?  3  from  5  ?  2  from 
6  ?    4  from  6  ?    3  from  6  ?     1  from  6  ?     5  from  6  ? 


MENTAL  AND    WRITTEN  ARITHMETIC. 


21 


LESSON      IX. 

L  One,  Two,  Three,  &c.,  are  called  Numbers  ;  and 

because  the  figures  1,  2,  3,  &c.,  stand  for  these  numbers, 
they  are  themselves  commonly  called  numbers. 

^.  When  we  put  2  and  3  together,  or  unite  them,  and 
find  that  they  equal  5,  we  are  said  to  Add  them. 

3,  Uniting  two  or  more  numbers,  and  finding  what 
number  they  equal,  when  taken  together,  is  named 
Addition, 

Jf,  When  numbers  are  to  be  added,  we  usually  write 
them  in  a  vertical  column.  Let  us  add  2,  3,  and  1. 
Writing  the  numbers  as  shown  in  the  margin,  and  .  2 
drawing  a  line  below  the  column,  we  first  find  that  3 
1  and  3  more  equal  4  N^ext  we  add  this  4  and  the  _ 
remaining  number,  2,  and,  finding  that  4  and  2  more  6 
equal  6,  we  write  6  below  the  line.  6  is  named 
the  Amount^  or  Sum^  of  2,  3,  and  1. 

Find  and  write  the  Sum  in  each  of  the  following 

Exercises  for  the  Slate  and  Blackboard. 
1321413222213 
212.4      13       2321131 
1111121122312 


23 


FIKST  LESSONS  IN 


LESSON   X, 

1.  Wlien  we  take  2  from  5  and  find  that  3  remain, 
or  are  left,  we  are  said  to  Subtract  2  from  5.  Subtract 
means  take  away, 

2.  Taking  one  number  from  another,  and  finding 
how  many  remain,  is  named  Subtraction. 

3.  Since  5  is  diminished,  or  made  less,  by  subtracting 
2  from  it,  we  name  5  the  Minuend.  Minuend  means 
to  be  diminished. 

Jf,  Since  2  is  subtracted  from  5,  2  is  named  the  Sub- 
trahend.     Subtrahend  means  to  be  subtracted. 

5.  Since  3  shows  how  many  remain  after  subtracting 
2  from  5,  we  name  3  the  JEteinainder^  or  the  Dlf^ 
ference  between  5  and  2. 

6.  In  performing  Subtraction  we 
usually  write  the  work  in  the  form 
shown  at  the  right  hand. 


SUBTRACTION. 


5  Minuend. 

2  Subtrahend. 

3  Remainder, 


Exercises  for  the  Slate  akd  Blackboard. 


5      3 
3      1 

2 


4 
2 


6      6 
2      3 


MENTAL  AND    WRITTEN  ARITHMETIC,  23 

LESSON  XI. 

2  +  3  =  5. 

i.  The  Sign  of  Equality  divides  every  Equation  into 
two  parts,  named  Memhei's. 

2,  The  First  Member  and  the  Second  Member  of  every 
Equation  are  equal,  and  the  Sign  of  Equality  stands  be- 
tween them. 

3,  Since  the  Equation  2  +  3  =  5  is  formed  by  Addi- 
tion, we  name  it  an  JEquation  by  Addition. 

4-  In  every  Equation  by  Addition  like  the  above,  hav- 
ing three  numbers,  with  the  greatest  standing  last,  if 
only  one  of  the  numbers  be  missing  it  is  easy  to  find  it. 

Since  the  Sum  of  the  first  two  numbers  equals  the 
third,  it  is  evident  that  if  either  of  them  be  subtracted 
from  the  third  the  Eemainder  will  equal  the  other 
number.  >fc 

We  may  find  any  one  of  the  three  numbers  thus : 

Missing  Numbers.  Methods  of  Finding. 

First.  Subtract  the  Seco:n'd  from  the  Third. 

Second.        Subtract  the  First  from  the  Third. 
Third.  Add  the  First  and  Secon^d. 

Write  the  proper  numbers  in  place  of  (?)  in  these 
Exercises  for  the  Slate  and  Board. 


2  +  3  =  ? 

3  +  1  =? 

?  +  1  =  5 

?  +  3  =  4 

2  +  2  =  ? 

3  +  2  =  ? 

4  +  ?  =  5 

2  +  ?  =  3 

1+4  =  ? 

3  +  ?  =  5 

2  +  ?  =  4 

1  +  ?  =3 

1  +  3  =  ? 

?   +  3  =  5 

3  +  ?  =  4 

?  +  2  =  3 

2  +  1  =  ? 

1  +  ?  =5 

?  +  2  =  4 

2    +   4r=? 

2  +  ?  =5 

?  +  4  =  5 

?  +  1  =  4 

3  +  ?  =  6 

3  +  3  =  ? 

4  +  ?  =  6 

1  +  ?  =  6 

?  +  4  =  6 

24  FIRST  LESSONS  IN 

LESSON  XII. 

•      5-2  =  3.. 

-/.  Since  the  Equation  5  -  2  =  3  is  formed  by  Sub- 
traction, we  will  name  it  an  Equation  by  Sub- 
traction. 

2,  In  this,  and  in  every  Equation  by  Subtraction 
having  only  three  numbers,  and  the  largest  of  the  num- 
bers standing  first,  the  first  number  is  equal  to  the  Sum 
of  the  two  others. 

S,  If  the  second  number  be  subtracted  from  the  first, 
the  Difference  will  equal  the  third ;  and  if  the  third  be 
subtracted  from  the  first,  the  Difference  will  equal  the 
second. 

From  this  it  is  evident  that  in  any  such  Equation  by 
Subtraction,  we  may  find  any  one  of  the  numbers  thus : 

Missing  Numbers.  Methods  of  Finding. 

EiRST.  Add  the  Second  and  Third. 

Second.         Subtract  the  Third  from  the  First. 
Third.  Subtract  the  Second  from  the  First. 

Find  and  write  the  missing  numbers  in  the  following 

Exercises  for  the  Slate  and  Board. 

5-2  =  ?  5-?  =  4  ?  -  1  ==  3  S-l=? 

5_4=,?  ?_3  =  2  4-1  =  ?  3-?=l 

5-?=:3  ?-2  =  2  4-3  =  ?  3-2  =  ? 

5-?=l  ?-2=3  4-2=?  3-?=2 

5-3=?  ?-4=l  4-?=2  6-2=? 

5-1=?  ?-l=4  4-?=l  6-?=3 

5_?^2  ?-3=l  4-?=3  ?-2=4 

6-3  =  ?  6-4  =  ?  ?-3=:3  6-?=4 

?-4=2  6-1=?  6-5=?  6-?=2 


MENTAL  AND    WRITTEN  ARITHMETIC. 


25 


UPPER  COUNTERS. 


1 

L 

3 

4 

5 

1 

2 

3 

4 

5 

_. 

^m 

^ 

1 

2 

3 

4 

^ 

k^^ 

^^ 

1^^ 

^=n-H 

1 

2 

3 

4 

5 

6 

2 

3 

4 

5 

6 

3 

4 

5 

6 

4 

5 

6 

5 

^ 

^ 

^ 

LESSON  XIII. 

We  now  proceed  to  make  a  Table,  on  the  plan  shown 
above.  The  Table  will  help  ns  in  adding  and  sub- 
tracting, in  performing  the  work  in  all  Examples  such 
as  we  have  had. 

First,  we  arrange,  side  by  side,  5  rows  of  little  cubes, 
with  5  cubes  in  each  row.  Then  we  place  another 
row  of  5  cubes  a  little  above  these,  and  a  like  row  a 
little  to  the  left  of  the  5  rows.  We  number  and  write 
on  the  cubes  of  the  last  two  rows,  "  1,  2,  3,  4,  5,^'  as 
shown.  These  two  rows,  thus  numbered,  are  to  be  used 
as  Counters. 

Suppose  we  wish  to  add  2  and  3  more,  and  write 
the  Sum  in  the  Table.  We  take  from  the  Lower  Count- 
ers the  cubes  numbered  "1,  2,"  and  from  the  Upper 
Counters  the  cubes  numbered  "1,  2,  3,"  and,  adding 
them,  find  their  Sum  is  5.  We  then  write  this  Sum  on 
the  cube  which  stands  at  the  right  of  the  cube  num- 
bered "2"  in  the  Lower  Counters,  and  below  the  cube 
numbered  "  3 ''  in  the  Upper  Counters. 

In  the  same  manner  we  find  and  write  the  Sum  of 
any  other  two  numbers  written  on  the  Counters. 


26  FIRST  LESSOJVS  IN 

The  Table  is  filled  as  far  as  6.    We  read  it  thus : 
1  and  1  are  2,  2  and  1  are  3,  3  and  2  are  5, 

1  and  2  are  3,  2  and  2  are  4,  3  and  3  are  6 ; 

1  and  3  are  4,  2  and  3  are  5,  4  and  1  are  5, 

1  and  4  are  5,  2  and  4  are  6 ;  4  and  2  are  6  ; 

1  and  5  are  6 ;  3  and  1  are  4,  5  and  1  are  6. 

We  will  name  this  an  Addition  Table.  We  may 
also  use  it  as  a  Subtraction  Table. 

If  we  add  2  and  3,  their  Stc7n  is  5 ;  as  appears  in  the 
Table.  If  we  Subtract  2  from  5,  the  Difference  is  3. 
We  obtain  this  result  from  the  Table,  thus :  1st,  we  find 
in  the  Lower  Counters  2,  which  is  to  be  subtracted ;  2d, 
we  pass  from  this  2  along  to  the  right,  and  find  5,  from 
which  2  is  to  be  subtracted ;  3d,  directly  above  this  5, 
in  the  Upper  Counters,  we  find  the  3,  which  is  the  Dif- 
ference between  2  and  5. 

We  read  this  as  a  Subtraction  Table  thus : 

1  from  2  leaves  1,  2  from  3  leave  1,  3  from  5  leave  2, 

1  from  3  leaves  2,  2  from  4  leave  2,  3  from  6  leave  3 ; 

1  from  4  leaves  3,  2  from  5  leave  3,  4  from  5  leave  1, 

1  from  5  leaves  4,  2  from  6  leave  4 ;  4  from  6  leave  2 ; 

1  from  6  leaves  5 ;  3  from  4  leave  1,  5  from  6  leave  1. 

Exercises  eor  the  Slate  and  Boaed. 

2+3=?  l+?=6  4-1=?  3+?=6 

24-4  =  ?  4  +  ?  =  6  3-2  =  ?  1  +  4=? 

3_1=^?  2+?=6  1  +  3  =  ?  5  +  ?  =  6 

4  +  ?::^5  14.5:=?  5__9^2  4-2=? 

Writtei^  Exercises.  i 

1,  Frank  had  4  peaches,  and  Henry  2;  how  many 
had  both  boys  ? 

2,  Emma  had  6  pinks,  and  gave  3  of  them  to  Walter; 
how  many  had  she  left  ? 


MENTAL  AND    WRITTEN  ARITHMETIC, 


27 


LESSON  XIV. 


By  Objects.        | 

3l    13 

r-. 

>A-  e 

sR  02 

- 

*E  ^^'! 

If 

20     02 

^a^i 

i 

\=\-  6 

4-    -4 

J  Q^ 

By  Mgtires. 


3-3 


-        4H 


-6 


2=2 


>=6 


-6 


We  will  now 
add  3,  2,  4,  2,  3 
and  4,  as  shown  at 
the  right  hand, 
and  lind  how 
many  times  we 
can  make  the 
number  6  from 
them. 

Beginning  at 
the  bottom,  the 
first  number  is  4. 
From  the  3  next 
above  this  we  take 

2,  which,  added  to 
4,  make  6.  Hav- 
ing taken  2  from 

3,  we  have  1  left. 
We  now  add  this 
1  to  the  2  stand- 
ing above  the  3,  and  have  3  for  the  Sum.  Taking  3  of 
the  4  ones  next  above,  and  adding  them  to  this  Sum,  3, 
we  have  6.  [N'ext  we  add  the  1,  left  from  4,  to  the  2 
standing  above  the  4,  and  find  that  their  Sum  is  3. 
Adding  together  this  Sum,  3,  and  the  last  number,  3, 
we  find  their  Sum  to  be  6.  Thus  we  make  the  number 
6  three  times  from  the  whole  column.  We  write  each  6 
in  the  Sum,  and  write  +  between  the  6's. 

While  performing  the  work  we  say  thus :  4  and  2  are 
6 ;  2  from  3  leave  1,  1  and  2  are  3,  3  and  3  are  6 ;  3  from 
4  leave  1,  1  and  2  are  3,  3  and  3  are  6.     The  entire  col- 
umn is  equal  to  3  times  6. 
3 


6+6+6.     Sum. 
3  times  6 


28  FIRST  LESSONS  IN 

In  the  same  manner  perform  the  work  in  each  of 
these 

Examples  tor  the  Slate  akd  Board. 


1 

4 

2 

2 

3 

3 

5 

4 

4 

1 

5 

4 

5 

4 

1 

3 

2 

5 

5 

5 

4 

3 

3 

5 

5 

4 

2 

4 

2 

2 

4 

2 

5 

4 

3 

5 

2 

5 

5 

3 

3 

4 

4 

3 

1 

5 

4 

3 

4 

3 

3 

3 

5 

5 

3 

4 

2 

4 

5 

4 

2 

3 

4 

3 

4 

3 

2 

4 

• 

4 

1 

3 

5 

4 

3 

2 

5 

4 

3 

LESSON  XV. 

1.  In  this  scnool,  one  boy  has  6  books  on  his  desk, 
and  another  has  1 ;  how  many  books  have  both  on  their 
desks  ?     6  books  and  1  book  are  how  many  ? 

2,  We  make  a  figure  Seven^  thns :  7.    Make  one. 
S,  How  many  caps  are  hanging  in  the  upper  row? 

How  many  in  the  lower  row  ?     How  many  caps  are 
there  in  all  ?     5  caps  and  2  caps  are  how  many  ? 

4.  How  many  bojs  are  standing?  How  many  are 
sitting  ?  How  many  boys  are  there  in  all  ?  4  boys  and 
3  boys  are  how  many  ? 

5.  How  many  girls  in  this  school?  If  1  of  them 
should  go  home,  how  many  would  be  left  ?  1  girl  from 
7  girls  leaves  how  many  ? 

6.  If  2  of  the  7  boys  should  go  and  take  their  caps 
and  go  home,  how  many  boys  would  be  left  ?  2  boys 
from  7  boys  leave  how  many  ?  2  caps  from  7  caps  leave 
how  many  ? 

7.  How  many  girls  are  standing?  If  3  of  them 
should  sit  down,  how  many  would  be  left  standing  ? 


MENTAL  AND   WRITTEN  ARITHMETIC, 


29 


If  the  girls  so  left  standing  should  go  home,  how 
many  girls  would  be  left  ?  4  girls  from  7  girls  leave 
how  many  ?    3  girls  from  7  leave  how  many  ? 

8,  How  many  books  are  on  the  teacher's  desk  ?  If  5 
of  them  should  be  taken  away,  how  many  would  be  left  ? 
5  books  from  7  leave  how  many  ? 

Wkitten"  Exercises. 

1.  James  had  3  marbles,  and  Harry  gave  him  4  more ; 
how  many  had  he  then  ?  He  lost  2  marbles ;  how  many 
had  he  left  ? 

2.  Charlie  had  2  peaches,  and  his  mother  gave  him  5 
more ;  how  many  had  he  in  all  ?  He  ate  4 ;  how  many 
had  he  remaining  ? 

3.  Walter  had  4  cents,  and  his  father,  gave  him  3 
more ;  how  many  had  he  then  ?  He  spent  3  cents  for  7 
plums;  how  many  were  left?  He  ate  5  plums;  how 
many  had  he  left  ? 


30 


FIRST  LESSONS  IN 


LESSON   XVL 


1.  Eecite  this  Table  as 
both  an  Addition  and  Sub- 
traction Table. 

2,  Write  the  following  in 
Equations,  and  read  them  : 

5  and  2  are  7  (5  +  2  =  7) 
4  and  3  are  7 
3  and  4  are  7 
2  and  5  are  7 
7,  less  4  equal  3; 
7  less  5  equal  2 ; 

6  less  3  equal  3 ; 


rPPER  COUNTERS. 


J_ 

J_ 

3. 

£ 

J^ 

^ 

1 

2 

3 

4 

5 

6 

7 

2 

3 

4 

5 

6 

7 

3 

4 

5 

6 

7 

4 

5 

6 

7 

5 

6 

7 

6 

^ 

_™ 





^ 

7  less  3  equal  4 ; 
5  less  2  equal  3 ; 


7  less  2  equal  5 ; 
5  less  3  equal  2. 


EXEECISES   FOE  THE   SlATE   AKD   BoARD. 
Jidditlon, 

222311251123 
132153214442 
322312311211 


SiibtracHon. 

6       6       7 
14       3 


LESSON   XVIL 

ci'D'DITIOJV  A.T   SIGHT, 

1.  If  I  write  letters,  thus,  ox,  dog,  horse,  you  can 
name  the  toords,  which  they  form,  at  first  sight,  without 
stopping  to  Sjjell  them. 

2    3    5 

2,  If  I  write  numters,  thus,  3,  4,  2,  you  may  become 

able  to  name  their  Sums,  without  stopping  to  add,  as 
readily  as  you  name  words. 


MENTAL  AND    WRITTEN  ARITHMETIC.  31 

S,  Copy  the  Exercises  on  your  slate,  and  name  the 
Sums,  going  from  the  left  to  the  right ;  tlien  from  right 
to  left;  and  linally  name  them  by  skipping,  in  every 
possible  manner.     Do  not  write  the  Sums. 

At  recitation  the  Exercises  will  be  written  on  the 
blackboard,  and  you  will  name ,  the  Sums  instantly,  as 
your  teacher  points  to  the  Exercises,  one  by  one. 

Addition  at  Sight, 

415233546223 
261534231322 

The  Sums  are  the  same  m  all  cases  wjiere  the  figures 
are  the  same,  though  the  order  of  the  figures  be  changed. 
Hence  it  is  necessary  only  that  we  l:now  what  figures 
we  have,  without  regarding  the  order  in  which  they 
stand. 

Write  the  missing  numbers  in  the  following 

Exercises  por  the  Slate  k^d  Board. 

5+?=:6  44-?  =  6  3+?  =  7  6+?=r7 

?-2=:5  4-3=?  5-4=?  ?_2  =  4 

2+?  =  7  6-5=:?  6-3=?  2  +  2=? 

5+?=7  4+?=7  ?+2=7  7-4=? 

5-2=?  5-3=?  6-2=?  7-?=2 

3+4=?  2+5=?  6-4=?  ?-4=3 

L  How  many  more  are  7  than  4  ?  Than  2  ?  6  ?  1  ? 
3?  5?  4? 

2,  How  many  less  are  2  than  7  ?  Than  5  ?  3  ?  6  ? 
4?  7? 

S,  How  many  less  are  3  than  5?4?7?6? 

Jf.  How  many  are  3  and  3  ?  2  and  5  ?  4  and  2  ? 
3  and  4  ? 


32  FIRST  LESSONS  IN 

LESSON   XVIIL 


BY  ADDITION. 

2  +  3=3^  a,id  3  +  2=^3. 


BY  SUBTRACTION. 
e> — 2 ^^3^    and   O — 3  =  2* 


1.  In  the  first  of  the  aboye  Equations,  2  +  3  is  the 
FiKST  Membee,  and  5  is  the  Second  Member.  The 
two  Members  of  an  Equation  are  always  equal,  and  the 
Sign  of  Equality  stands  between  them. 

2,  Since  the  Sum  of  2  and  3  is  5,  we  may  write  two 
Equations  by  Addition:  2  +  3  =  3^  and  3  +  2  =  3. 

S,  Again,  since  the  Sum  of  2  and  3  is  5,  it  is  evident 
that  if  2  be  subtracted  from  5,  the  Eemainder  will  be 
3 ;  and  that  if  3  be  subtracted  from  5,  the  Eemainder  will 
be  2.  Hence,  we  write  two  Equations  by  Subtraction : 
3  -  2  =  3,  SLTid  3  -  3  =  2. 

U*  Thus,  from  the  three  numbers,  2^  3^  and  3,  we 
have  formed  four  Equations :  2 -\- 3  =  3,  3  +  2  =  3; 
3-  2=3,  and  3 -3 --=2. 

5,  From  any  three  unequal  numbers,  such  that  the 
Sum  of  the  two  smaller  ones  equals  the  largest,  we  may 
form  tzvo  Equations  by  Addition  and  tivo  Equations  by 
Subtraction. 

Eule  I. 
To  Form  Equations  by  Addition: 

I. — For  the  First  Member  of  an  Equation,  ivrite  tlie 
ttuo  smaller  numbers  with  the  Sign  Plus  between  them. 

II. — For  the  Second  Member,  write  the  largest  of  the 
three  numbers,  placing  the  Sign  of  Equality  bettveen  the 
Members, 

III. — Form  the  second  Equation  by  Addition  from  the 
first,  by  changing  the  places  of  the  two  smaller  numbers. 


MENTAL   AND    WRITTEN  ARITHMETIC,  33 

EULE   11. 

To  Form  Equations  by  Subtraction: 

I. — For  the  First  Member  of  mi  Equation^  tvrite  the 
largest  of  the  three  numbers,  a?id  after  it  one  of  the  ttvo 
smaller  numbers,  ijlacing  the  Sig7i  Minus  between  them, 

II. — For  the  Second  Member,  ivrite  the  other  of  the  two 
smaller  numbers,  placing  the  Sign  of  Equality  between 
the  Members, 

III. — Form  the  second  Equation  by  Subtraction  from 
the  first,  by  changijig  the  places  of  tjie  two  smaller 
numbers. 

Note. — When  the  two  smaller  numbers  are  eqnal,  the  two 
Equations  by  Addition  will  be  precisely  alike,  and  also  those  by 
Subtraction. 

In  the  manner  directed  by  the  preceding  Kules,  form 
and  write  four  Equations  from  each  of  the  following 

Groups  of  Three  Numhers, 

1,  3  and  4 ;       1,  5  and  6  ;       1,  6  and  7 ;       2,  5  and  7 ; 

2,  4  and  6  ;       1,  4  and  5  ;       3,  4  and  7  ;       2,  3  and  5. 

If  the  three  numbers  are  given  in  an  Equation,  it  is 
plain  that  we  can  form  three  more  Equations  from  this. 

Form  three  other  Equations  from  each  of  the  fol- 
lowing 

Equations, 

44-2  =  6;     5  +  1  =  6;     7-3  =  4;     7-1  =  6; 
7-2  =  5;     6-1  =  5;    3+2  =  5;     4  +  1  =  5. 


Addition  at  Sight. 

3 

2 

3 

2 

1 
3 

3 
3 

4         3          3          1 

3       3       4       6 

5 

1 

3 
5 

3 
4 

4 
3 

34 


FIRST  LESSONS  IN 


LESSON   XIX. 

1,  In  this  picture,  how  many  peaches  are  on  Willie's 
table  ?  How  many  on  Mary's  table  ?  How  many  on 
both  ?     7  and  1  are  how  many  ? 

2,  How  many  oranges  are  on  Willie's  table  ?  How 
many  on  Mary's  ?  How  many  on  both  ?  6  oranges  and 
2  oranges  are  how  many  ? 

3,  How  many  pears  are  on  Mary's  table  ?  How  many 
on  Willie's  ?  How  many  on  both  ?  5  pears  and  3  pears 
are  how  many  ? 

Jf,  How  many  apples  are  on  Mary's  table?  How 
many  on  Willie's  ?  How  many  on  both  ?  •  4  apples  and 
4  apples  more  are  how  many  ? 

How  many  are 

1  and  7  ?  2  and  6  ?  2  and  3  ?  5  and  3  ?  4  and  4  ? 

2  and  1  ?  3  and  5  ?  2  and  4  ?  6  and  2  ?  3  and  4  ? 


MENTAL   AND    WRITTEN  ARITHMETIC.  35 

5.  If  Mary  should  give  Elizabeth  her  peach,  how 
many  peaches  would  she  and  Willie  have  left?  8 
peaches  less  1  peach  are  how  many  ? 

6,  If  WiUie  should  give  Harry  3  pears,  how  many 
would  he  and  Mary  have  left  ?  8  pears  less  3  pears  are 
how  many? 

7.  If  Mary  should  give  Jane  2  oranges,  how  many 
would  she  and  WiUie  have  left?  8  oranges  less  2 
oranges  are  how  many  ? 

8,  If  Willie  should  give  his  4  apples  to  his  mother, 
how  many  would  he  and  Mary  have  left  ?  8  apples  less 
4  apples  are  how  many  ? 

How  many  are 
8  less  2  ?  8  less  4  ?  8  less  6  ?  8  less  3  ? 

8  less  5?  8  less  7?  8  less  1  ?  7  less  4? 


LESSON   XX. 


Add  the  numbers  in  each  of  the  following  columns 
by  5's,  in  the  manner  explained  on  page  27,  for  adding 
by  6's : 


3 

3 

4 

3      4      4      2       3       1 

2 

3 

4 

4 

5 

3 

4      3       2       4      12 

2 

3 

4 

2 

1 

4 

5       2       2       14      5 

2 

3 

4 

2 

4 

1 

2       4       4       5       3       4 

2 

3 

4 

4 

2 

3 

12       3       3       4      3 

Add  the  following  by  6's 

- 

2 

3 

4 

5 

3 

4 

5       2       2       3       5       1 

2 

3 

4 

5 

4 

5 

3 

4      5       4      14      1 

2 

3 

4 

5 

2 

4 

2 

2       12      3      2       1 

2 

3 

4 

5 

3 

5 

4 

4      3       3      4      4      1 

2 

3 

4 

5 

5 

3 

2 

5      4      2      3      2       1 

2 

3 

4 

5 

5 

4 

3 

4      3      5      4      11 

2 

3 

4 

5 

36  FIRST  LESSONS  IN 

* 

WRiTTEi^  Exercises. 

1,  Walter  and  Albert  went  hunting.  Walter  shot  5 
squirrels,  and  Albert  3 ;  how  many  did  both  kill  ?  They 
lost  2  of  the  squirrels ;  how  many  were  left  ?  Walter 
killed  6  pigeons,  and  Albert  2 ;  how  many  did  both 
kill?  They  gave  away  4  of  the  pigeons;  how  many 
were  left  ? 

2,  Anna,  Amelia,  and  Willie  went  to  gather  flowers. 
Anna  picked  4  lilies,  and  Willie  gave  her  4  more  ;  how 
many  had  she  in  all  ?  She  lost  3 ;  how  many  had  she 
left?  Amelia  picked  7  lilies,  and  Willie  gave  her  1; 
how  many  had  she  then  ?  She  gave  her  mother  5  lilies ; 
how  many  had  she  at  last  ? 


LESSON  XXL 

Write  the  following  in  Equations,  and  read  them : 

7  and  1  are  8,  3  and  5  are  8,  7  less  6  equal  1 ; 

6  and  2  are  8,  2  and  6  are  8,  8  less  5  equal  3, 

5  and  3  are  8,  1  and  7  are  8 ;  8  less  7  equal  1, 
4  and  4  are  8,  7  less  3  equal  4,  8  less  4  equal  4, 
4  and  3  are  7 ;  7  less  5  equal  2,  8  less  6  equal  2, 

6  and  1  are  7,  7  less  1  equal  6,  8  less  3  equal  5, 

2  and  5  are  7,         7  less  2  equal  5,         8  less  1  equal  7, 

3  and  4  are  7,         7  less  4  equal  3,         8  less  2  equal  6. 
Write  four  Equations  from  each  of  the  following 

Groups  of  Three  Numbers. 

1,  7,  and  8 ;    2,  6,  and  8 ;    3,  5,  and  8  ;    2,  5,  and  7 
3,  4,  and  7. 

Write  three  others  from  each  of  the  following 

JEquations. 

3  +  6  =  8;  4  +  3  =  7;  5  +  3  =  8;  8-5=3; 
1  +  7  =  8;     8-3  =  6;    7-3  =  4;    3  +  5=7. 


mental  and  written  arithmetic,         37 
Exercises  for  the  Slate  akd  Board, 

Addition, 


I. 

1 

2 

2 

3 

2 

5 

2 

2 

1 

5 

1 

6 

4 

3 

3 

4 

3 

2 

2 

1 

6 

5 

1 

6 

1 

2 

4 

2 

2 

2 

4 

1 

5 
II. 

1 

2 

2 

1 

1 

2 

3 

2 

3 

4 

4 

4 

2 

3 

3 

3 

4 

1 

2 

1 

3 

2 

3 

1 

2 

4 

3 

2 

1 

1 

4 

1 

4 

3 

3 

1 

3 

1 

1 

1 

2 

3 

2 

3 

4 

Subtraction. 

- 

6 

4 

5 

3 

6 
3 

6 
2 

7 
3 

8      8      7 
3      5      3 

8 

2 

8 
4 

7 
4 

8 
6 

7 
5 

LESSON   XXI L 

BQZrA.TIOJ\rS. 


2  +  ?  =  8  5  +  ?  =  7  7-2  =  ?  4  +  ?  =r  8 

v+3==8  8-2  =  ?  7-?=4  3  +  ?=:8 

3+?=8  8 ~ ? = 5  ?-4=3  ?+4=8 

2  +  5  =  ?  ?  -  4  =  4  7-3  =  ?  7-4  =  ? 

3+?=7  8-5=?  7-5=?  ?-3=5 

?+4=7  ?+6=8  6+2=?  ?+2=8 

Add  the  numbers  in  the  following  columns  by  7's,  in 
the  manner  explained  on  page  27,  for  adding  by  6's : 


4 

5 

6 

5 

4 

2 

5 

4 

5 

3 

5 

4 

5 

5 

5 

3 

5 

5 

4 

1 

6 

6 

2 

6 

6 

4 

4 

3 

4 

2 

4 

4 

6 

6 

6 

6 

4 

3 

3 

3 

4 

5 

3 

6 

5 

4 

6 

6 

5 

5 

6 

5 

5 

4 

3 

6 

2 

6 

5 

6 

5 

5 

1 

2 

4 

38 


FISST  LESSONS  7iV 


LESSON   XXIII. 

1.  Learn   this  Addition 
and  Subtraction  Table. 

2.  If  we  write  the  Equa- 
tion 3  +  4  =  7  in  the  form 
of  an  Example  in  Addition, 
as  shown  at  the 
right,  the  order    3  )  p^^^^^ 
of  the  numbers     4  f 
is  not  changed.     7     Sum. 
Hence  we  may 
use  the  method  shown  on 
page  23  also   To  find  any 

Number  in  an  Example  in  Addition,  when  missing ; 
thus: 


_L 

2 

3 

£ 

A 

6 

7_ 

1 

2 

2 

3 

4- 

5 

6 

7 

8 

3 

4 

5 

6 

7 

8 

3 

4 

5 

6 

7 

8 

4 

5 

6 

7 

8 

5 

6 

7 

8 

6 

7 

8 

7 

± 

« 

^ 

^ 

Names. 


Methods  of  Finding. 

Subtract  the  Second  from 

the  Third. 
Subtract  the  First  from  the 

Third. 
Add  the  First  and  Second. 


Exercises  eor  the  Slate  and  Board. 


5 
3 

9 


? 

2 
7 


I. 
? 
5 


4 

7 


4 

9 


8      8 


? 
3 

7 


6       8 


II. 
? 

2 
6 


?       ? 

i     1 

8      4 


MENTAL  AND  WRITTEN  ARITHMETIC. 


39 


LESSON      XXIV. 

If  we  take  the  Equation  8  —  3=5,  and  write  it  in  the 


form  of  an  Example  in  Subtraction, 
as  shown  at  the  right,  the  order  of 
the  numbers  is  not  changed.  Hence 
we  may  use  the  method  shown  on 
page  24  also  To  find  a  missing 
numher  in  an  Example  in  Subtraction  ;  thus  : 


8  Minuend, 
3  Subtrahend. 

^  5  Difference, 


Missing  Numbers. 

Names. 

Methods  of  Finding. 

ElEST. 

MlKUEI^D. 

Add  the  Second  and 
Third. 

SECOi^D. 

SUBTRAHEN^D, 

Subtract  the  Third 
from  the  Eirst. 

Thikd. 

Difference. 

Subtract  the  Second 
from  the  First. 

Exercises  for  the 

Slate  and  Board. 

7       7       8 

1. 

8       8       ? 

? 

?       ?       ?       ?       ? 

3       4       3 

5       4       5 

2 

4       3       6       3       2 

?     ? 


?     ? 


II. 


7867?   778   ??86 
?   ?   3   ?   5   ?   ?   ?.   3   4   ?  .  ? 


In  the  same  manner  as  we  added  by  7's  on  page  37, 
add  each  column  by  8's  in  the  following 

Exercises  for  the  Slate  and  Board. 


7 

7 

6 

5 

4 

6 

7 

6 

6 

5 

4 

5 

5 

3 

6 

5 

7 

6 

5 

5 

6 

5 

3 

4 

3 

3 

3 

4 

4 

3 

2 

4 

3 

3 

6 

5 

0 

6 

7 

3 

6 

5 

3 

5 

4 

7 

4 

7 

5 

6 

2 

7 

3 

4 

7 

4 

5 

4 

7 

3 

40 


FIRST  LESSONS  IN 


LESSON   XXV. 

1,  In  this  picture,  how  many  beehives  in  each  of  the 
two  rows  ?  How  many  in  both  rows  ?  You  see  one 
hive  standing  apart  from  the  two  rows.  If  you  count 
this  with  the  others,  how  many  are  there  ?  We  make  a 
figure  Nine  thus:  9.  Make  one.  One  and  how 
many  more  make  9  ? 

2,  How  many  birds  do  you  see  oyer  this  house  ?  How 
many  are  about  to  light  on  the  tree  ?  How  many  birds 
in  all  ?     7  birds  and  2  more  are  how  many  ? 

|.  S,  How  many  squirrels  have  climbed  the  tree  ?  How 
many  others  are  running  towards  the  tree  ?  4  squirrels 
and  2  more  are  how  many  ?  How  many  squirrels  are 
on  the  fence,  running  away  from  the  tree  ?  How  many 
squirrels  in  all  ? 


MENTAL  AND    WRITTEN  ARITHMETIC,  41 

Jf,  How  many  windows  in  the  npper  story,  in  the  side 
of  the  house  ?  How  many  in  the  lower  story,  in  the 
side  of  the  house  ?  How  many  windows  are  3  and  2 
more  ?  How  many  windows  do  you  see  in  the  and  of 
the  house  ?  5  and  4  more  are  how  many  ?  If  a  door 
should  be  put  in  place  of  one  of  the  windows,  how  many 
would  be  left  ?     1  window  from  9  leaves  how  many? 

5,  If  the  boy  shoot  two  of  the  squirrels,  how  many 
will  be  left  ?     2  from  9  leave  how  many  ? 

6,  If  3  hives  be  carried  away,  how  many  will  remain  ? 
3  from  9  leave  how  many  ? 

7,  K  the  boy  shoot  4  of  the  birds,  how  many  will  be 
left  ?    4  from  9  leave  how  many  ? 

Weittek  Exercises. 

1,  Harry  and  Edward  went  fishing.  Harry  caught  2 
sunfish,  and  Edward  7  ;  how  many  did  both  catch  ?  2 
and  7  more  are  how  many  ?  3  of  the  fishes  were  lost 
fi-om  the  basket ;  how  many  were  left  ?  3  from  9  leave 
how  many? 

2,  Harry  caught  3  perch,  and  Edward  6  ;  how  many 
did  both  boys  catch  ?  3  and  6  are  how  many  ?  They 
gave  away  5  perch  ;  how  many  were  left  ?  5  perch  from 
9  leave  how  many  ? 

3,  Harry  caught  5  bass,  and  Edward  4 ;  how  many 
were  caught  in  all?  5  and  4  are  how  many?  They 
gave  away  2  bass ;  how  many  were  left  ?  2  from  9  leave 
how  many  ?  They  also  lost  2  bass.  4  bass  from  9  leave 
how  many  ? 

Jf,  Harry  caught  6  eels,  and  lost  4  of  them;  how 
many  had  he  left  ?  He  then  caught  2  more  ;  how  many 
had  he  at  last  ?  Edward  caught  5  eels ;  how  many  eels 
had  Harry  and  Edward,  to  carry  home  ?  4  eels  and  5 
eels  are  how  many  ? 


42  FIRST  LESSONS  IN 

LESSON  XXVI. 

EXEEOISES  FOR  THE   SlATE  A.^I>   BoAED. 

AddUion. 

I. 

6  4  523235537262 
23344142231214 

II. 

3262543323343 
231/2111242111 
3112222121234 
1212112312111 


Subtraction, 

9 
3 

7 
3 

8 
3 

7 
3 

9 

3 

8      7      9      8 
4      4      5      2 

7      9 
5.^6 

8 
5 

9 

7 

9 
4 

JEquations, 

7-f?=9  ?+2  =  9  9-3  =  ?  9-2  =  ? 

9 -_  5  =  ?  ?-3r=6  9-?=5  9  -  ?  =  3 

4+?=9  9-?=2  ?-5=4  ?-4=5 

54-?=9  3  +  ?  =  9  9  -  ?  =:=  4  9-?  =  6 

Examples  in  JLddition, 

3735??4212?2??4?? 

6?     ??     24?     7?     ?     3?     ^    ^    'i    I    ^ 
9989998?"989998998 

Examples  in  Subtraction. 

9998  9.98??9?9??9?8 
62??4??78?6?54?2? 

3?73?  5  4213324  4  472 


MENTAL  AND    WRITTEN  ARITHMETIC, 


43 


LESSON  XXVII. 

1.  Frank  had  2  apples  on  a  fniit-dish.  He  took  2 
apples  from  the  dish  and  gave  them  to  a  beggar.  How 
many  apples  were  left  on  the  dish?  2  apples  from  2 
apples  leave  how  many  apples  ?  We  use  a  figure  written 
thus,  0^  to  stand  for  Nothing.  Its  name  is  Zero^ 
or  Naughty  or  Cipher.  Each  of  these  words  means 
KoTHiKG ;  and  the  figure,  0^  also  means  Nothing, 

2,  In  Arithmetic  we  use  no  figures  but  these :  0,  1, 
2,  3,  4,  5,  6,  7,  8,  9. 

Write  the  following  in  Equations;  and  from  each 
Equation  so  written  write  three  others  in  the  manner 
explained  in  Lesson  XVIII. 


5  and  4  are  9, 

6  and  3  are  9  ; 

9  less  3  equal  6, 
9  less  6  equal  3  ; 
4 


7  and  2  are  9, 
2  and  7  are  9 ; 
9  less  5  equal  4, 
9  less  4  equal  5  ; 


4  and  5  are  9, 
3  and  6  are  9 ; 
9  less  2  equal  7, 
9  less  7  equal  2. 


44 


FIRST  LESSONS  IN 


Learn  and  recite 

tte  annexed  Addi- 

tion 

and   Subtrac- 

tion  Table : 

Add  the  nunabers 

in 

the 

following  ( 

30l- 

umns 

by9's 

6 

3 

7    6 

4 

4 

5 

6 

8    6 

5 

2 

7 

7 

2    4 

6 

5 

3 

5 

5     8 

6 

7 

5 

4 

5     5 

5 

3 

3 

6 

6     3 

8 

7 

7 

5 

3     4 

% 

8 

J_ 

Z^ 

A 

£ 

i 

1. 

L 

8^ 

1 

2 

3 

4. 

5 

6 

7 

8 

9 

2 

3 

4 

5 

6 

7 

8 

9 

3 

4. 

5 

6 

7 

8 

9 

4 

5 

6 

7 

8 

9 

5 

6 

7 

8 

9 

6 

7 

8 

9 

7 

8 

9 

8 

9 



^ 

„, 

ee* 

_ 

^ 

^ 

LESSON  XXVIII. 

1,  In  the  cut  on  the  next  page,  in  the  first  row,  how 
many  cubes  in  the  greater  part?  How  many  in  the 
other  part  ?  How  many,  in  all,  in  that  row  ?  9  cubes 
and  1  more  are  how  many  ? 

2,  In  the  second  row,  how  many  cubes  in  the  greater 
part  ?  How  many  in  the  other  part  ?  How  many,  in 
all,  in  the  second  row  ?    8  cubes  and  2  more  are  how 


many 


?    2  and  8  more  ? 


S,  In  the  third  row,  how  many  cubes  in  the  greater 
part  ?  In  the  other  part  ?  In  the  third  row  ?  7  cubes 
and  3  more  are  how  many  ?     3  and  7  more  ? 

^.  In  the  fourth  row,  how  many  cubes  in  the  greater 
part?  In  the  other?  In  the  fourth  row?  6  cubes 
and  4  more  are  how  many  ?     4  and  6  more  ? 

5.  In  the  fifth  row,  how  many  cubes  in  each  part? 
5  cubes  and  5  more  are  how  many  ? 


MENTAL  AND    WRITTEN  ARITHMETIC. 


45 


mm. 


3/4-/5 


/a/4-  /6  /e , 


nETHROW 
FOURTH  BOW 
TEQBD  HOW 
SECOND  ROW 
FIRST    ROW 


6.  If  we  take  1  cube  from  the  first  row,  how  many 
will  be  left?  If,  instead,  we  take  away '9  cubes,  how 
many  will  remain  ?  1  cube  from  Ten  cubes  leaves  how 
many  ?    9  cubes  from  Ten"  cubes  leave  how  many  ? 

7.  How  many  cubes  will  be  left  in  the  second  row,  if 
we  take  away  2  cubes  ?  If  we  take  8  ?  2  cubes  from 
Te]^  leave  how  many  ?    8  from  Tek  how  many  ? 

8.  How  many  cubes  will  be  left  in  the  third  row,  if 
we  take  away  3  ?  If  we  take  7  ?  3  cubes  from  Teit 
leave  how  many  ?     7  from  TEiq"  how  many  ? 

9.  Four  cubes  taken  from  the  fourth  row  will  leave 
how  many  ?  6  taken  away  leave  how  many  ?  4  cubes 
from  Tek  leave  how  many  ?     6  from  Te^  how  many  ? 

10.  Five  cubes  taken  from  the  fifth  row  will  leave 
how  many  ?    5  cubes  taken  from  Ten  leave  how  many  ? 

Learn  the  following 


TA-HILE: 


9  and  1  are  Ten^, 
8  and  2  are  Ten", 
7  and  3  are  Teist, 
6  and  4  are  Ten, 
5  and  5  are  Ten, 
4  and  6  are  Ten, 
3  and  7  are  Ten, 
2  and  8  are  Ten, 
1  and  9  are  Ten  ; 


1  from  Ten  leaves  9, 

2  from  Ten  leave  8, 

3  from  Ten  leave  7, 

4  from  Ten  leave  6, 

5  from  Ten  leave  5, 

6  from  Ten  leave  4, 

7  from  Ten  leave  3, 

8  from  Ten  leave  2, 

9  from  Ten  leave  1. 


46  FIRST  LESSONS  IN 


^0!««Ki 


LESSON  XXIX. 

L  Walter  arranged  his  set  of  Alphabet  Blocks  on  a 
sheet  of  paper,  in  the  order  shown  above.  He  then  pro- 
ceeded to  count  them,  printing  the  numbers  on  the 
paper  in  words,  just  above  the  blocks.  You  see  that 
there  were  Twenty-six  blocks. 

2,  He  counted  them  again,  and  wrote  the  numbers 
just  below  the  blocks,  in  figures.  For  the  blocks  A,  B, 
0,  D,  E,  F,  G,  H,  I,  he  counted  and  wrote  1,  2,  3,  4,  5, 
6,  7,  8,  9.  For  the  block  J  he  counted  Ten,  as  before ; 
but,  not  finding  any  single  figure  for  Tee",  he  printed 
the  ivord  TEiT.  Counting  the  blocks  K,  L,  M,  N,  0,  P, 
Q,  E,  S,  T,  he  wrote  below  them  1,  2,  3,  4,  5,  6,  7,  8,  9, 
Teit. 

For  the  blocks  IT,  V,  W,  X,  Y,  Z,  he  counted  and 
wrote  1,  2,  3,  4,  5,  6. 

Finding  that  the  two  sets  of  numbers  were  unlike 
beyond  Tek,  and  not  knowing  how  to  write  a  number 


MENTAL  AND    WRITTEN  ARITHMETIC. 


47 


greater  than  Nine,  in  figures,  he  was  at  first  puzzled. 
Finally,  he  wrote  and  gave  to  his  teacher  the  following 


TEN 

and  1 
and  2 
and  3 
and  4 
and  5 
and  6 
and  7 
and  8 
and  9 


TABLE. 

2  TENS 


are  Eleven, 
are  Twelve, 
are  Thirteen, 
are  Fourteen, 
are  Fifteen, 
are  Sixteen, 
are  Seventeen, 
are  Eighteen, 
are  Nineteen. 


are  Twenty, 
and  1  are  Twenty-one, 
and  2  are  Twenty-two, 
and  3  are  Twenty-three, 
and  4  are  Twenty-four, 
and  5  are  Twenty-five, 
and  6  are  Twenty-six. 


LESSON   XXX. 


myinnMEiMniEMMiiuMWM^M 


flRST    TEN-CROU P 


EaEaEiEOiimmmmBxaamvm^m'm 


«3   W 
H  O 


SECOND    TEN-GROUP  26 

TWEKTY-SIX. 


SIX  ONES 


Mary  counted  and  numbered  Tei^  of  her  Alphabet 
Blocks,  and,  naming  them  a  Ten-group^  placed  them 
at  her  left  hand.  Forming  a  second  Ten-group,  she 
placed  it  with  the  first.  The  remaining  Six  blocks  she 
numbered  and  placed  at  the  right. 

Said  she  to  her  teacher :  "  I  have  Six  single  blocks, 
standing  at  the  right.  I  will  write  a  figure  6,  to  stand 
for  these.  Since  I  have  Two  Ten-groups,  I  will  writs  a 
figure  2,  to  stand  for  them ;  remembering  that  this  2 


48 


FIRST  LESSONS  IN 


does  not  stand  for  2  blocTcs^  but  for  2  Ten-groups ^  each 
having  Ten  blocks. 

"  Since  the  2  Ten-groups  are  placed  at  the  left  of  the 
6  single  blocks,  I  will  write  the  figure  2,  which  stands  for 
the  2  Ten-groups,  at  the  left  of  the  /^?^re  6,  which 
stands  for  the  6  single  blocks ;  thus,  20." 

"  That  is  correct/^  answered  the  teacher.  "  Two  Tens 
are  named  Twenty.  .  Your  figures,  26,  are  to  be  read 
Twentg-six,  For  an]^  number  of  Ones,  or  single  blocks 
which  are  not  more  than  9,  you  write  one  figure.  Then, 
if  you  have  any  number  of  Tens  which  are  not  more 
than  9,  you  write  one  figure  for  them ;  placing  it  at  the 
left  of  the  figure  which  stands  for  the  Ones»  If  there 
are  no  Ones,  you  write  a  Cipher ,  thus,  0,  at  the  right  of 
the  figure  standing  for  the  Tens.  78  equals  7  Tens  and 
8  Ones ;  90  equals  9  Tens  and  no  (0)  Ones." 


Some  numbers  are  named  as  shown  in  the  following 
tabz:e: 


Thirty, 
^  Forty, 


3  Tens 

4  Tens 

5  Tens  |  Fifty, 

6  Tens  |  Sixty, 

7  Tens  ^  Seventy, 

8  Tens  '^  Eighty, 

9  Tens     Ninety. 


Thirty-one,  etc. 
.  ^  Forty-one,  etc. 


3  Tens  and  1  One 

4  Tens  and  1  One  ^ ^  ._.,... 

5  Tens  and  1  One  |  Fifty-one,  etc. 

6  Tens  and  1  One  |  Sixty-one,  etc. 

7  Tens  and  1  One  |^  Sevenfcy-one,etc. 

8  Tens  and  1  One  ®  Eighty-one,  etc. 

9  Tens  and  1  One     Ninety-one,  etc. 


Some  numbers  greater  than  9  are  written  thus : 

TABLE. 


By  Words.   By  Figures. 

Ten,  10; 

Eleven,  11 : 

Twelve,  12 : 

Thirteen,  13: 


By  Words.    By  Figures. 

Fourteen,  14 ; 

Fifteen,  15 

Sixteen,  16 

Seventeen,  17 


By  Words.  By  Figures. 

Eighteen,  18 
Nineteen,  19 
Twenty,  20 
Twenty-one,  21 


MENTAL  AND    WRITTEN  ARITHMETIC, 


49 


By  Words.     By  Figures.    By  Words,  By  Figures.  By  Words.  By  Figures. 

Twenty-two,     22;  Thirty-one,  31;  Seventy,         70 

Twenty-three,  23 ;  Forty,  40 ;  Eighty,  80 

Twenty-four,    24;  Fifty,  50;  Ninety,  90 

Thirty,  30;  Sixty,  60;  Ninety-nine,  99. 


LESSON    XXXI. 

TVrlie  the  following  J^umbers  hi  JP'lgures . 


Twenty-seven ; 
Thirty-three ; 
Fifteen ; 
Thirty-nine ; 
Fourteen ; 
Forty- five; 
Thirteen ; 


Fifty-three ; 
Eleven ; 
Sixty-eight ; 
Twenty ; 
Seventy-seven ; 
Thirty; 
Eighty-six ; 


'Forty; 
Thirty-four ; 
Seventeen ; 
Nineteen ; 
Ninety-three ; 
Fifty; 
Ninety-nine. 


Bead  aloud  the  following  Numbers : 

17        41        50        64        55        36  69  58 

27        44        13        12        91        10  9  84 

37        49        72        85        11        93  47  39 

Exercises  for  the  Slate  and  Board. 

Addition. 


60 
80 
99 


3 

4 

2 

5 

2 

2 

3 

2 

4 

3 

4 

6 

4 

4 

2 

3 

1 

2 

1 

5 

4 

3 

1 

3 

1 

5 

3 

3 

5 

4 

5 

7 

2 

4 

3 

6 

2 

3 

1 

Subtraction. 

9 

10 

8 

10 

10      10       9 

10 

10 

9 

9 

2 

% 

4 

5 

3         6       5 

4 

J_ 

3 

4 

JEquations. 

5  +  5  =  ?         5  +  ?  =  10         10-5  =  ?  ?-5  =  5 

?+4  =  10       6  +  ?  =  10        10-?=4  ?-3  =  l. 

10-?=:3         ?--7  =  3             ?+2  =  10  ?-2  =  8 


50 


MBST  LESSONS  IN 


LESSON   XXXII. 

1.  Learn  and  re- 
cite the  annexed 
Table : 

2,  2  Ones  and  3 
Ones  are  5  Ones; 
and  2  Tens  and  3 
Tens  are  5  Tens,  or 
50.  3  Ones  taken 
from  7  Ones  leave 
4  Ones  ;  and  3  Tens 
taken  from  7  J'e/xs 
leave  4  I'e^^,  or  40. 

In     every    case, 
Tens  are  added  to 
jT^/i^,  or  subtracted  from  Tens,  in  the  same  manner  as 
Ones  are  added  to  or  subtracted  from  Ones. 


1 

2_ 

^ 

£^ 

± 

6 

7 

8 

^ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

To 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

3 

4. 

5 

6 

7 

8 

9 

10 

II 

4 

5 

6 

7 

8 

9 

10 

II 

5 

6 

7 

8 

9 

10 

II 

6 

7 

8 

9 

10 

II 

7 

8 

9 

10 

il 

8 

9 

10 

II 

. 

9 

10 

11 

^ 

,_. 

Exercises  for  the  Slate  and  Board. 

JLddition, 

I. 


11 
12 

21 
13 

32 
23 

43 
34 

49 
30 

54 
22 

37 
52 

65 
31 

46 
23 

81 
17 

76 
13 

21 
14 
32 

15 
32 
40 

30 
24 
15 

51 
16 
21 

11 
23 
45 

n. 

33 

23 
11 

12 
23 
34 

42 
13 
24 

52 
24 
12 

16 
63 
20 

71 
14 
14 

Subtraction. 

• 

22 

11 

33 
22 

45 
33 

52 
20 

58 
33 

65 
41 

87 
32 

76 
45 

69 
35 

88 
73 

99 
33 

MENTAL  AND    WRITTEN  ARITHMETIC.  51 

LESSON   XXXIII. 

Mental  Exeecises. 


9  +  3  =  ? 

?  +  4  =  ll 

8  +  3  =  ? 

?  +  5  =  ll 

8  +  ?  =  ii 

6  +  ?  =  11 

3  +  ?  =  11 

7  +  ?  =  ll 

6  +  5  =  ?  ■ 

?  +  8  =  ll 

?  +  7  =  11 

?  +  6  =  11 

3  +  ?  =  11 

5  +  ?  =11 

?  +  2  =  ll 

4  +  ?  =  11 

?  +  3  =  11 

11  -  2  =  ? 

?  +  9  =  ll 

7  +  4  =  ? 

11  -  4  =  ? 

11-6  =  ? 

11  -  3  =  ? 

11-5  =  ? 

11-8  =  ? 

11  -  1  =  ? 

11-7  =  ?  ' 

11-9  =  ? 

10  -  3  =  ? 

10  -  5  =  ? 

10  -  6  =  ? 

10  -  4  =  ? 

Exercises  for  the  Slate  ai^d  Board. 

1,  Frank  had  4  rabbits,  and  his  father  gave  him  7 
more ;  how  many  had  he  in  all  ? 

2,  A  farmer  liad  16  sheep  in  one  pasture,  and  22  in 
another ;  how  many  had  he  in  both  ?  He  sold  25  sheep ; 
how  many  had  he  left  ? 

3,  In  a  school  there  were  43  boys,  and  35  girls ;  how 
many  pupils  in  all?  15  pupils  left;  how  many  re- 
mained ? 

^.  Willie  found  a  hen's  nest  with  15  eggs,  and  Walter 
found  one  with  13  ;  how  many  eggs  did  both  find? 

5.  In  one  flock  of  pigeons  were  54,  and  in  another 
35  ;  how  many  pigeons  in  both  flocks  ? 

Addition, 


52 
23 

34 

15 

18 
21 

26 
-4i 

57   84 
31   13 

Subtraction, 

45 
23 

74 
22 

56 
23 

64 
14 

74 
31 

98 
35 

76 
43 

69 
25 

48   87 
15   36 

36 

14 

66 
33 

52 
40 

75 
25 

53 


FIRST  LESSONS  IN 


LESSON   XXXIV. 


A/k\/ih,v/,tf,u\J:if^'h///i 


1,  In  the  upper  row  of  these  cubes,  how  many  cubes 
are  in  the  greater  part  ?  How  many  in  the  other  part  ? 
In  both  parts  ? 

2,  In  the  lower  row,  how  many  cubes  are  in  the 
greater  part  ?    In  the  other  part  ?    In  both  ? 

S,  We  wish  to  find  how  many  cubes  there  are  in  all; 
that  is,  find  the  Sum  of  18  and  17. 

1st.  We  first  add  the  7  cubes  in  the  lower  row,  and  the 
8  in  the  upper  row.  Since  7  and  3  are  10,  we  take  3  of 
the  8  cubes,  and,  adding  them  to  the  7,  have  10  cubes. 
Since  3  taken  from  8  leave  5,  we  have  5  of  the  8  cubes 
left. 

As  shown  in  the  following  cut, 


we  separate  the  8  cubes  into  3  cubes  and  5  cubes,  and 
then  place  the  3  cubes  at  the  end  of  the  row  of  7  cubes. 
Counting  them  together,  we  have  10  cubes. 

Thus  we  have  in  all  a  group  of  10  cubes,  and  5  single 
cubes ;  which  are  15  cubes. 

2d.  We  next  arrange  the  cubes  as  shown  in  the  cut 
at  the  top  of  the  opposite  page.  The  Ten-group,  which 
we  have  formed,  we  carry  to  the  left  and  place  with 


MENTAL  AND    WRITTEN  ARITHMETIC.  53 


the  2  other  Ten-groups.  Counting  the  2  Tens  with  the 
1  Ten  CARRIED,  we  find  they  are  3  Tens.  Thus,  we 
have,  in  all,  3  Tens  and  5  Ones;  which  are  35  cubes. 

In  like  manner,  we  add  18  and  17  by 
figures.   We  write  the  numbers  as  shown    addition. 
at  the  right.     Taking  3  of  the  8  Ones,        18  ]  p^rts. 
we  add  them  to  the  7  Ones  to  make  10,        ^  i 
and  then  add  the  remaining  5  Ones  to        35     Sum. 
the  10,  and  thus  obtain  15  as  the  Sum 
of  8  and  7.    We  then  carry  to  the  left  the  1  Ten  of 
the  15,  and,  addmg  it  with  the  1  Ten  of  the  17  and  the 
1  Ten  of  the  18,  obtain  3  Tens.     Thus  we  find  that  the 
Sum  of  18  and  17  is  3  Tens  and  5  Ones,  or  35. 

In  like  manner  we  add  any  hvo  or  more  numhers, 
when  their  Sum  is  not  greater  than  99.  We  carry  to 
THE  left  all  the  Tens  formed  by  adding  the  figures  in 
the  right-hand  column. 

TBSTIJV-G  THB  SUM. 

After  adding  numbers  we  sometimes  doubt  the  cor- 
rectness of  our  work.  In  such  cases  it  is  well  to  add 
the  figures  a  second  time,  commencing  at  the  top  and 
adding  to  the  bottom.  If  we  obtain  the  same  result  as 
in  the  first  instance,  it  is  presumed  that  we  have  found 
the  true  Sum.  This  second  addition  is  named  Test' 
ing  the  Sum. 


54 


FIRST  LESSONS  IN 


LESSON   XXXV, 


This  Table  should 
be  learned  so  thor- 
oughly that  any 
part  of  it  can  be 
given  without  hesi- 
tation. Give  special 
attention  to  that 
part  of  the  Table 
which  includes 
numbers  greater 
than  10.  No  fur- 
ther Table  for  Ad- 
dition or  Subtrac- 
tion will  be  needed 
if  this  be  mastered. 


I 

2 

3 

4 

5 

6 

7 

\1. 

\1 

^grf^^s 

i?g=^I^^g 

ImmI 

lgs=s^ 

1 

^^m 

1 

2 

2 

3 

4 

5 

6 

7 

8 

9 

10 

3 

4 

5 

6 

7 

8 

9 

10 

II 

3 
4 

4 

5 

6 

7 

8 

9 

10 

11 

12 

5 

6 

7 

8 

9 

10 

II 

12 

13 

5 

6 

7 

8 

9 

10 

II 

12 

13 

14 

6 

7 

8 

9 

10 

II 

12 

13 

14 

15 

7 
8 

8 

9 

10 

II 

12 

13 

14 

15 

16 

9 

10 

II 

12 

13 

14 

15 

16 

17 

9 

10 

II 

12 

14 

15 

16 

17 

18 

Exercises  eor  the  Slate  akd  Board. 


Addition, 


25 

65 


16 
34 


37 

13 


48 
12 


56 
35 


27 
16 


57 
26 


46 
37 


54 

29 


II. 

8 

23 

39 

63 

18 

19 

24 

11 

62 

42 

7 

50 

23 

14 

25 

44 

18 

45 

18 

19 

5 

17 

28 

19 

34 

23 

HI. 

35 

37 

16 

27 

9 

14 

16 

12 

34 

12 

11 

14 

13 

19 

6 

35 

13 

14 

13 

14 

14 

31 

27 

33 

8 

13 

47 

17 

28 

36 

42 

14 

18 

22 

7 

28 

14 

33 

13 

27 

19 

28 

30 

11 

MENTAL  AND    WRITTEN  ARITHMETIC. 


55 


LESSON   XXXVI. 


Add  by  Ti. 

Add  by  8'e. 

AddbyO's.  , 

Add  by  lO'a. 

Add. 

6     3     1 

7           6 

7           8 

8        7 

15         18 

5     6     6 

.5           7 

8           7 

9        8 

34        14 

4    3     5 

6         5 

6         5 

5        6 

10        33 

5     4    4 

3        7 

3         6 

6         5 

37        17 

16     5 

4        7 

4        1 

3         4 

14        35 

Written  Exercises. 

1.  How  many  bushels  of  apples  in  3  piles,  the  first 
containing  26  bushels,  the  second  35  bushels,  and  the 
third  29  bushels? 

2,  A  poultry  dealer  sold  some  turkeys  for  46  dollars, 
some  geese  for  23  dollars,  and  some  chickens  for  27  dol-  . 
lars ;  how  many  dollars  did  he  receive  in  all  ? 

S.  A  farmer  sold  38  bushels  of  wheat,  26  bushels  of 
corn,  17  bushels  of  rye,  and  16  bushels  of  barley;  how 
many  bushels  of  grain  did  he  sell  in  all  ? 

Jf,  In  3  days,  Albert  picked  23  quarts  of  strawberries, 
Harry  29  quarts,  Alfred  34  quarts,  and  Walter  10  quarts; 
how  many  quarts  did  the  4  boys  pick  ? 

5,  How  many  pigeons  are  twenty-eight  pigeons, 
thirty-four  pigeons,  and  twenty-nine  pigeons  ? 

6,  Frank  rode  twenty-nine  miles  Monday,  thirty 
Tuesday,  and  eighteen  Wednesday;  how  many  miles 
did  he  ride  during  the  three  days? 

7,  Willie  had  47  peaches  in  his  basket,  Frederick  16 
in  his,  and  Henry  28  in  his ;  how  many  peaches  had  the 
three  boys  ? 

8,  Homer  had  45  oranges  in  his  basket,  and  Horatio 
39  in  his ;  how  many  oranges  had  both  boys  ? 

9,  Nellie  spelled  39  words,  and  Mary  48 ;  how  many 
did  both  spell? 


56 


FIRST  LESSONS  IN 


LESSON  XXXVIL 

1.— MINUEND. 


■■iHiHiE3EHE3fli 

■ 

■•  'mM 

WHnifliEiOI^M 

lilil|ii'iii::^::^vr:i:.^;i^^^ 

iii 


SUBTRAHEND. 


1 

IT 

E 

:n'         I 

■ 

Iii 

1 

i 

i!!i:'- 

J! 

I 

The  riglit-liand  figure  in  the  Subtrahend  greater  than 
the  figure  above  it  in  the  Minuend, 

Example.    From  35  subtract  18.  subtraction. 

Writing  the  18  under  the  35,  we  find       35  Minuend, 
that  8  cannot  be  subtracted  from  5.  18  Subtrahend, 

First :  Subtraction  hij  Objects, 

The  above  cut  is  in  2  Parts.  In  Part  1.  how 
many  Ten-groups  are  in  the  Minuend  ?  How  many 
single  cubes  ?  3  Tens  and  5  Ones  are  how  many  ? 
How  many  Ten-groups  are  in  the  Subtrahend  ?  How 
many  single  cubes  ?     1  Ten  and  8  Ones  are  how  many  ? 


MENTAL  AND    WRITTEN  ARITHMETIC,  57 

This  Minuend,  then,  having  35  cubes,  and  Subtrahend 
having  18  cubes,  answer  to  the  Minuend  and  Subtrahend 
in  the  given  Example.  Hence  the  Difference  between 
35  cubes  and  18  cubes  will  be  the  answer  to  the  given 
Example. 

Since  8  cubes  cannot  be  subtracted  from  5  cubes,  we 
take  1  of  the  3  Ten-groups  in  the  Minuend  and  caery 
it  TO  THE  RIGHT,  and  place  it  with  the  5  cubes,  as  10 
separate  cubes. 

This  Minuend,  as  re-arranged,  is  shown'  in  Part  11.  of 
the  cut.  Since  the  Minuend  and  Subtrahend  in  Part  11. 
are  equal  to  those  in  Part  I.,  the  Difference  in  Part  II. 
and  in  Part  I.  must  be  the  same.  We  will  find  the  Dif- 
ference in  Part  II. 

We  separate  the  Ten-group,  which  stands  with  the  5 
cubes,  into  two  parts,  one  having  8  cubes  and  the  other 
2.  Since  there  are  8  cubes  in  the  Subtrahend,  we  sub- 
tract 8  cubes  from  the  10  cubes.  There  are  then  left, 
in  the  Minuend,  2  of  the  10  cubes  and  also  the  5  cubes. 
Placing  5  cubes  and  2  cubes  in  the  Difference,  we  have 
7  cubes  as  the  Difference  between  the  8  cubes  of  the 
Subtrahend  and  the  10  cubes  and  5  cubes  of  the 
Minuend. 

In  Part  I.  we  had  1  Ten  to  subtract  from  3  Tens ;  but 
in  Part  IL,  having  carried  to  the  right  1  Ten  in  the 
Minuend,  we  have  only  2  Tens  remaining  at  the  left. 
Subtracting  1  Ten  from  2  Tens,  and  obtaining  1  Ten 
for  the  Difference,  we  place  1  Ten-group  in  the  Differ- 
ence. Thus  we  find  that  18  cubes  taken  from  35  cubes 
leave  17  cubes. 

Second  :  Subtraction  by  Figures, 

Wliat  we  have  done  with  the  cubes  we  will  now  do 
with  the  figures.    As  shown  in  the  margin,  we  write 


58  FIRST  LESSONS  IN 

the  18  under  the  35.    We  then  re-arrange  the  Minuend, 
by  first  taking  1  of  the  3  Tens  and 

CAERYIKG  it  TO  THE  RIGHT  and  writ-  Subtraction. 

ing  it  as  10  Ones,  over  the  5  Ones,      !.  ^^   ,^. 
and  then  drawing  a  Hne  through  the       f  I  Minuend, 
3,  to  show  that  it  is  not  to  be  used,      ^  SuUrahend. 
and  writing  the  2  remaining  Tens      ^  ^  Difference, 
over  the  former  3. 

First,  we  subtract  8  from  5  and  10  taken  together. 
Subtracting,  8  from  10  leave  2 ;  and  this  2  and  the  5, 
added  together,  give  7  Ones  for  the  Difference.  Sub- 
tracting 1  Ten  from  2  Tens,  we  have  1  Ten  for  the 
Difference  in  Tens.  Thus  we  obtain  for  our  Difference 
1  Ten  and  7  Ones ;  or  17. 

In  every  Example  like  this,  the  work  is  performed  in 
the  manner  just  explained.     It  is  not  necessary,  how- 
ever, to  change  the  figures  of  the  Minuend.     In  this 
Exam^^le,  we  write  the  numbers  as 
shown  in  the  margin.     Subtracting,  subtraction. 

we  say,  not  8  from  5,  but  8  from  10       35  Minuend, 
leave  2 ;   2  and  5  are  7 ;   and  then      18  SuUrahend, 
write  7  in  the  Difference.     Finally,       17  Difference. 
we  say,  not  1  Ten  from  3  Tens,  but  1 
Ten  from  2  Tens  leaves  1  Ten ;  and  write  1  Ten  in  the 
Difference. 


Examples 

POR  • 

THE  Slate  xi^jy  Board. 

29 
13 

54 
29 

35 

18 

72 
36 

I. 

91 
43 

II. 

57 
29 

74 
68 

33 
16 

96 
48 

65 
43 

82 
37 

23 

15 

44 

28 

63 

27 

56 

28 

71 
58 

62 
33 

78 
39 

MENTAL  AND    WRITTEN  ARITHMETIC.  69 

LESSON   XXXVIII. 
su:bt^a  ctiojv. 

Testing  tlie  Difference, 

If  the  Difference  between  two  numbers  be  added  to 
the  less,  the  Sum  will  equal  the  greater.  If  this  Differ- 
ence be  subtracted  from  the  greater  number,  the  result 
will  equal  the  less.  Hence,  we  may  Test  -our  Difference 
in  Subtraction  by  either  of  two  methods. 

First  Method:  Testing  by  Addition, 

Add  the  Difference  to  the  Subtrahend,  If  the  Sum 
equals  the  Minuend,  it  is  presumed  that  we  have  found 
the  true  Difference. 

Second  Method:   Testing  hy  Stibtraction. 

Subtract  the  Difference  from  the  Minuend.  If  the  re- 
sult equals  the  first  Subtrahend  it  is  presumed  that  the 
first  Dfference  is  correct. 

Find  and  test  the  Difference  in  each  of  the  following 
Examples  for  the  Slate  akd  Boaed. 

Testing  by  First  Method, 


83 

52 

70        93        80       47 

90 

75        65 

64 

38 

36        79        39        18 

Testing  lyy  Second,  MethoA. 

45 

25       47 

33 

79 

43         53         39        62 

71 

80       91 

19 

49 

28         14        19        53 

AdeUHan  at  Sight. 

52 

65        74 

5 

7      9 

6       4       8       4      5       9 

3 

5       5      6 

4 

4      4 

4      4      4      5       5       5 

5 

6       8      3 

60  FIRST  LESSONS  IJf 

LESSON  XXXIX. 


Addition, 

u 

19 

24 

18 

37 

19 

23 

13 

12 

15 

23 

n 

32 

14 

25 

11 

41 

29 

16 

16 

15 

13 

13 

13 

23 

13 

29 

17 

28 

17 

29 

28 

17 

35 

32 

29 

Subtraction. 

83 

94 

64 

77 

85 

97 

54 

78 

91 

57 

31 

24 

59 

25 

88 

27 

59 

42 

Written  Exercises. 

Addition  and  Subtraction. 

L  Edwin  had  a  set  of  26  Alphabet  Blocks  on  his 
desk,  and  Susan  had  the  same  number  on  her  desk ; 
how  many  blocks  did  both  have  ? 

Edwin  put  17  of  his  blocks  in  the  box;  how  many 
were  left  on  his  desk  ? 

Susan  put  19  of  her  blocks  in  the  box ;  how  many 
were  left  on  her  desk  ? 

How  many  did  both  put  in  their  boxes  ? 

How  many  remained  on  both  desks  ? 

2.  Mr.  Newton  had  55  sheep,  and  Mr.  Lawton  had 
42 ;  how  many  sheep  did  both  have  ? 

How  many  more  sheep  had  Mr.  Newton  than  Mr. 
Lawton  ? 

Mr.  Newton  sold  29  sheep  to  Mr.  Lawton ;  how  many 
had  Mr.  Lawton  then  ? 

How  many  had  Mr.  Newton  left  ? 

How  many  more  had  Mr.  Lawton  than  Mr.  Newton  ? 


Addition  at  Sight. 

7 
5 

9 

6 

7 
6 

8 
6 

6       5       9       7      4 

g      Y      7      7      7 

8 

7 

6 
9 

6 

7 

6 
8 

MENTAL  AND    WRITTEN  ARITHMETIC,  61 

LESSON  XL 
Examples  for  the  Slate  and  Board. 

Addition. 


18 

27 

13 

16 

15 

16 

17 

18 

19 

13 

13 

28 

23 

15 

16 

17 

18 

19 

17 

14 

12 

11 

15 

16 

17 

18 

19 

18 

25 

17 

35 

15 

16 

17 

18 

19 

19 

16 

25 

13 

15 

16 

17 

18 

19 

Mental  Exercises. 

1,  One  of  2  hens  has  7  chickens,  and  the  other  9 ; 
how  many  chickens  have  both  hens  ? 

2,  Charles  has  9  marbles,  and  Frank  6 ;  how  many 
marbles  have  both  boys? 

5,  Flora  picked' 8  quarts  of  strawberries,  and  Ella  9; 
how  many  quarts  did  both  pick  ? 

Jf.  Edward  shot  11  squirrels,  and  Henry  6  ;  how  many 
more  did  Edward  shoot  than  Henry  ? 
How  many  did  both  boys  shoot? 

6,  Harry  bought  12  peaches,  and  gave  5  of  them  to 
his  sister ;  how  many  had  he  left  ? 

6,  Walter  was  18  years  old,  and  "Willie  9;  how  many 
years  was  Walter  older  than  Willie  ? 

7.  On  a  tree  were  17  pigeons,  and  9  of  them  flew 
away ;  how  many  were  left  on  the  tree  ? 

Addition  at  Sight* 


2 
8 

7 
4 

3 

8 

5 

7 

8      7      4      8      5 
5      7      8      6      8 

Subtraction  at  Sight. 

8 

7 

9 

5 

8 
8 

3 
9 

4 

2 

5 
3 

5 
_3 

6 
3 

6      6      7      7      8 
2      4      2      4      3 

7 
3 

7 
5 

8 
4 

8 
6 

62 


FIRST  LESSONS  IN 


One  iluNDKED  and        Fifty-  Six; 

or,  in  figures, 

156, 

LESSON   XL/. 

A  maker  of  Alphabet  Blocks  prepared  6  sets  for  let- 
tering, and  requested  his  little  daughter  to  count  them 
and  tell  him  how  many  there  were.  She  counted  them, 
wrote  on  them,  and  arranged  them  as  shown  in  the 
above  cut. 

Said  she  to  her  father:  "Because  I  cannot  count 
more  than  Ten,  I  have  counted  the  blocks  in  groups, 
each  having  Ten  blocks.  I  have  placed  each  Ten  in  a 
row,  writing  the  numbers  on  the  blocks  as  T  counted 
them,  and  finally  arranged  the  groups  side  by  side. 
After  making  as  many  Tens  as  I  could,  I  placed  the 
remaining  blocks  at  the  right  of  these,  and  counted 
them  1,  2,  3,  4,  5,  6. 

"  I  have  tried  to  count  my  Ten-groups,  but  find  there 
are  more  than  Ten  of  them.  "When,  at  first,  I  was 
counting  the  single  Uoclcs,  I  counted  them  in  groups  of 
Ten  single  Uoclcs,  because  I  could  not  count  more  than 
Ten.  Now,  also,  when  I  am  counting  my  Ten-groupSy 
because  I  cannot  count  beyond  Ten,  I  will,  in  the  same 


MENTAL  AND    WRITTEN  ARITHMETIC,  63 

manner,  count  my  Ten-groups  into  larger  groups,  each 
having  Ten  Ten-groups,  As  I  count  these  Ten-groups 
I  number  them,  and  write  on  the  ends  of  them  1,  2,  3, 
4,  5,  6,  7,  8,  9,  Ten.  These  Ten  Ten -groups  I  separate 
from  the  others,  and  remove  them  a  little  to  the  left, 
and  call  them  1  large  group.  I  count  the  other  Ten- 
groups,  and  write  on  them  1,  2,  3,  4,  5. 

"  Since  there  are  six  single  blocks  at  the  right,  I  will 
write  a  figure  6,  to  stand  for  them ;  thus,  6.  Since 
there  SiYefive  Ten-groups  standing  by  themselves,  at  the 
left  of  the  six  blocks,  I  will  write  a  figure  5  to  stand  for 
them ;  placing  it  at  the  left  of  the  figure  6.  Since  there 
is  one  large  group,  at  the  left  of  the  5  Ten-groups,  I  will 
write  a  figure  1  to  stand  for  this ;  placing  it  at  the  left 
of  the  figure  5, 

"  My  figures  will  then  be  placed  in  the  same  order  as 
the  groups  and  single  blocks ;  thus,  1  5  6.^' 

Her  father  said :  "  Your  large  group  is  named  One 
Hundred ;  your  5  Ten-groups  are  named  Fifty ;  and 
your  figures,  156,  are  read  One  Hundred  and  Fifty-sixP 

Numbers  greater  than  99  are  named  and  written  thus : 

TAJiljE. 

10  Tens  are  One  Hundred ;  written,  100. 
20  Tens  are  Two  Hundred ;  written,  200. 
30  Tens  are  Three  Hundred ;  written,  300. 
40  Tens  are  Four  Hundred ;  written,  400. 
60  Tens  are  Five  Hundred ;  written,  500. 
60  Tens  are  Six  Hundred ;  written,  600. 
70  Tens  are  Seven  Hundred ;  written,  700. 
80  Tens  are  Eight  Hundred  ;  written,  800. 
90  Tens  are  Nine   Hundred ;  written,  900. 

101  is  read  One   Hundred  and  One. 
570  is  read  Five  Hundred  and  Seventy. 
999  is  read  Nine  Hundred  and  Ninety-Nine. 


64  FIRST  LESSONS  IN 

LESSON   XLII. 

Write,  in  figures,  the  following  : 

Five  Hundred  and  Twenty-  Nine  Hundred  and  Ninety- 
three  ;  nine ; 

Four  Hundred  and  Eighty-  Seven  Hundred  and  Eleven; 
seven ; 

Six  Hundred  and  Ninety;  Five  Hundred  and  Seven; 

Eight  Hundred  and  Forty ;  Seven  Hundred ; 

Seven  Hundred  and  Ten ;  Three  Hundred  and  Eight; 

Addition. 

Example.  Find  the  Sum  of  456  and  231. 

ExPLA:srATiON.  We  add  the  Ones  and  Tens  in  the 
same  manner  as  if  there  were  no  Hun- 
dreds, and  finding  the  Sum  to  be  87  addition. 
write  it  under  the  columns.    Adding        456  |  p^^^^ 
2  Hundreds  and  4  Hundreds  in  the        ^^  3 
same  manner  as  we  added  Ones,  and        687     Sum, 
Tens,  and  writing  the  Sum,  6  Hundreds, 
we  have  687  as  the  entire  Sum  of  456  and  231. 

Exercises  for  the  Slate  ai^d  Board. 
I. 


147 
236 

238 
354 

546 
234 

329 
453 

176 
215 

415 
127 

824 
159 

213 
158 

215 
326 
144 

423 
135 

218 

154 
218 
323 

II. 
231 
154 

208 

119 
218 
351 

235 
144 
216 

310 

207 
156 

407 
200 
126 

Siihtraction, 

We  subtract  Hundreds  from  Hundreds  in  the  same 
manner  as  we  do  Tens  from  Tens,  or  Ones  from  Ones. 

Exercises  for  the  Slate  akd  Board. 
549        763        452        985        691       826       398       745 
321        245        147        756        243       518       289       245 


MENTAL  AND    WRITTEN  ARITRMETia  65 

LESSON   XLin. 

Carrying  every  10  Tens  to  the  Left  as  1  Hundred. 

Example. — Find  the  Sum  of  574  and  253. 

ExPLAi^ATiOiq^. — Adding  the   Ones,  we  write  their 
Sum,  7.    Adding  the  Tens,  their  Sum  is 
12  Tens;  or  10  Tens  and  2  Tens.    We    addition. 
write  the  2  Tens  under  the  column  of      574 )  p^^^^^ 
Tens.    Since  the  10  Tens  are  1  Hun-      ?^^ ) 
dred,  we  carry  them  to  the  left  as  1  Hun-      827    Sum, 
dred,  and  add  this  Hundred  with  the 
other  Hundreds,  in  the  same  manner  as  heretofore,  in 
adding  Tens  and  Ones,  we  have  carried  to  the  left  every 
10  Ones,  from  the  column  of  Ones,  and  added  them  as 
1  Ten  with  the  other  Tens  at  the  left.    Adding  2  Hun- 
dreds, 5  Hundreds,  and  the  I  Hmidred  which  we  carried 
to  the  left,  we  write  8  under  the  column  of  Hundreds. 

Thus  we  find  the  Sum  of  574  and  253  to  be  827. 

Exercises  eor  the  Slate  akd  Board. 


235 

162 

421 

L 

324 

142 

170 

214 

362 

142 

231 

235 

132 

250 

218 

152 

123 

231 

354 

162 

261 

134 

221 

451 

154 

152 

215 

315 

IL 

173 

219  ' 

123 

324 

218 

237 

143 

267 

326 

102 

257 

153 

172 

123 

374 

184 

184 

391 

164 

247 

154 

132 

125 

212 

115 
III 

173 

231 

141 

431 

173 

214 

365 

143 

237 

378 

215 

132 

254 

175 

138 

375 

114 

152 

171 

269 

135 

143 

243 

114 

345 

125 

124 

171 

156 

257 

156 

252 

103 

261 

453 

183 

66  FIRST  LESSONS  IN 

LESSON   XLIV. 

Carrying  1  Hundred  to  the  Right  as  10  Tens. 

Example.  Subtract  379  from  652. 

Expla:n-atio:n'.  Carrying  to  the  right  1  of  the  5  Tens 
in  the  Minuend,  and  subtracting  9 
Ones  from  10  Ones  and  2  Ones,  solution. 

we  have  3  Ones  left.     Hence  we  f^^  Minuend. 

.,    o;i^i,       1  ^rk  379  Subtrahend. 

write  3  under  the  column  oi  Ones.  — -- 

Having  carried  to  the  right  1  of  ^^^  Difference. 
the  5  Tens  in  the  Minuend,  there  are  only  4  Tens  left. 
Since  we  can  not  subtract  7  Tens  from  4  Tens,  we  carry 
to  the  right  1  of  the  6  Hundreds  in  the  Minuend,  and, 
calling  it  10  Tens,  subtract  the  7  Tens  from  the  10  Tens 
and  4  Tens,  saying:  7  Tens  from  10  Tens  leave  3  Tens; 
3  Tens  and  4  Tens  are  7  Tens.    We  then  write  7  Tens. 

Having  carried  to  the  right  1  of  the  6  Hundreds  in 
the  Minuend,  we  now  subtract  the  3  Hundreds  in  the 
Subtrahend  from  the  5  Hundreds  left  in  the  Minuend, 
and  write  2  Hundreds  in  the  Difference. 

Hence  the  Difference  between  379  and  652  is  273. 


EXEKCISES  : 

FOE  THE  Slate 

A.^jy   BOAED. 

576 
234 

765 
284 

648 

257 

I. 

429   946 
156   493 

II. 

328 
254 

724 
251 

514 
127 

842 

257 

578 
259 

421 
156 

627   724 
253   468 

III. 

218 
104 

327 
139 

876 
588 

711 
524 

526 
389 

410 
215 

674   836 

285   647 

325 
147 

939 

756 

575 
197 

MENTAL  AND    WRITTEN  ARITHMETIC.  67 


LESSON   XLV. 

Exercises  for  the  Slate  ais^d  Board. 

Addition, 


158 

189 

176 

184   176 

154 

187 

197 

176 

147 

124 

117   145 

238 

176 

137 

167 

135 

189 

269   179 

197 

198 

167 

134 

178 

165 

175   186 

286 

184 

147 

179 

186 

248 

236   179  , 
II. 

124 

153 

157 

123 

134 

125 

136   137 

148 

129 

148 

123 

134 

135 

116   117 

128 

119 

128 

123 

134 

125 

126   147 

118 

139 

118 

123 

134 

115 

116   127 

138 

159 

138 

123 

134 

135 

136   117 

158. 

119 

158 

123 

134 

115 

126   137 

138 

139 

118 

123 

134 

125 

136   117 

128 

149 

121 

Addition  at  Sight, 

9   7   4 

8   2 

6   9   9 

9   9 

5   9   9 

9   9   9 

9   9 

9   4   8 

2   6 

9   7   3 

Subtraction, 

524 

743 

621 

I. 

812   546 

954 

726 

314 

173 

429 

238 

734   268 
II. 

267 

459 

198 

957 

742 

525 

326   813 

640 

435 

293 

243 

529 

389 

178   508 

239 

276 

108 

Subtraction  at  Sight 

, 

9   1 

)   8 

9   8 

10   9   10 

9 

8   10 

10 

2   4   3 

3   5 

_5   6   _8 

5 

6   _6 

4 

68 


FIRST  LESSONS  IJST 


ONE   'j.kJ.<j^i:,.i^±j  ONE  HUNDRED  AND  ELBYBN; 

or,  written  in  figures, 

1,111. 

LESSON   XLVI. 

A  mechanic  sawed  out  a  basketful  of  Alphabet  Blocks 
and  gave  them  to  his  little  son  to  count. 

The  boy  first  counted  them  and  arranged  them  as 
shown  in  the  above  cut.  Then,  turning  to  his  father, 
he  said:  '^I  cannot  count  beyond  Ten.  I  first  counted 
the  blocks  in  groups  having  Ten  blocks  in  each.  Each 
of  these  groups  I  call  a  Ten-groujp.  After  making  as 
many  Ten-groups  as  I  could  I  had  1  single  block  left, 
which  I  have  placed  by  itself,  at  the  right  hand. 

"  I  next  counted  these  Ten-groups  in  the  same  man- 
ner as  I  counted  the  single  blocks ;  putting  Ten  Ten- 


HENTAL  AND  WRITTEN  ABITIIMETIC.  G9 

groups  together,  and  placing  them  one  Ten-group  above 
another ;  thus  making  of  every  Ten  Ten-groups  1  larger 
group. 

"  After  making  as  many  as  I  could  of  these  larger 
groups,  I  had  1  Ten-group  remaining,  which  I  have 
placed  at  the  left  of  the  single  block." 

"  Each  of  these  larger  groups  is  named  a  Hundred  ; 
and  the  1  Ten-group  and  the-  single  block,  taken  to- 
gether, are  named  eleven,"  remarked  the-  father. 

"  Then,"  said  the  son,  "  I  will  call  each  of  these  larger 
groups  a  Hui^DRED-GROUP.  I  then  counted  my  Hun- 
dred-groups, placing  Ten  of  them  side  by  side,  to  make 
1  very  large  group.  I  had  1  Hundred-group  remaining, 
which  I  placed  at  the  left  of  the  1  Ten-group.  At  the 
left  of  this  Hundred-group  I  placed  the  1  very  large 
group." 

Said  the  father:  "This  largest  group  is  named  a 
Thousand.  Can  you  tell  me  how  many  blocks  there 
are,  and  then  write  the  number  h^  figures  V^ 

The  son,  after  reflecting  a  moment,  promptly  replied : 
"There  are  Oxe  Thousan^d  Oke  Hundred  and 
Eleven  blocks.  I  have  one  single  block,  standing  alone, 
and  also  one  group  of  each  kind  standing  by  itself.  I 
will  write  a  figure  1  to  stand  for  the  one  block,  and  also 
a  figure  1  to  stand  for  each  one  group  of  the  different 
kinds;  writing  the  figures  in  the  same  order  as  the 
groups  are  placed ;  thus,  1111." 

The  boy  answered  correctly.  And  always  in  writing 
a  number  having  Thousands,  we  write  a  figure  showing 
the  number  of  Thousands,  placing  it  at  the  left  of  the 
figure  written  for  Hundreds. 

Eead  the  following  numbers : 

1111;  1119;  1110;  1211;  1201;  1200;  1000;  5009; 
5900;  5990;  7080;  5017;  1001;  9999. 


W  FIRST  LESSONS  IN 

LESSON   XLVIL 

JVO  TA  no  AT  ;;ij[r^  JSrUMB'UA  TIOJV. 

Writing  numbers  in  figures  is  named  Notation. 

When  numbers  are  written  in  figures,  reading  them 
in  luords  is  named  Numeration* 

Eead,  or  Numerate,  the  following  numbers : 

1560,  1506,  1056,  156,  1500,  1050,  1005,  7089,  9090, 
9017,  1921,  5786,  1870. 

Write  in  figures  the  following  numbers : 

1,  Three  Thousand  Five  Hundred  and  Seventy-six ; 

2,  One  Thousand  Two  Hundred  and  Ten ; 
S,  Two  Thousand  One  Hundred  and  Three ; 
^.  Four  Thousand  and  Thirty ; 

5.  Five  Thousand  and  Thirteen ; 

6.  Seven  Thousand  and  Three ; 

7.  One  Thousand  Two  Hundred. 

A.ddition, 

In  adding  Hundreds  and  Thousands,  we  carry  to  the 
left  every  10  Hundreds,  calling  them  1  Thousand,  and 
add  this  Thousand  with  those  in  the  column  of  Thou- 
sands. 

Exercises  eor  the  Slate  ai^d  Board. 


1542 

1020 

2153 

1450 

1000 

2157 

1740 

2176 

1507 

1208 

1234 

2000 

1528 

2031 

1628 

2418 

1759 

2357 

2060 

1S72 

1507 

3473 

1256 

1080 

1567 

1007 

2143 

1423 

1024 

2569 

3417 

1728 

3000 

1726 

2158 

A.ddition  at  Sight, 

8   5 

9   6 

7   9 

7   7 

8 

8   8 

8   9 

5   9 

6   8 

6   7 

7   8 

6 

8   9 

7   9 

MENTAL  AND    WRITTEN  ABITHMETIC,  71 

LESSON  XLVIII. 
su:B2'^actiojv. 

In  Subtraction,  whenever  the  Hundreds  in  the  Sub- 
trahend are  more  than  those  above  them  in  the  Minuend, 
we  carry  to  the  right  1  of  the  Thousands  in  the  Minuend, 
calhng  it  10  Hundreds,  in  the  same  manner  as  we  have 
heretofore  carried  to  the  right  1  Hundred,  calhng  it  10 
Tens. 

EXEBCISES  FOR  THE   SlATE  AifD   BOAED. 


5497 
2145 

7421 
5176 

9354 

5927 

I. 

3257 
1476 

8542 
4716 

6215 
4389 

9666 
5999 

4571 

2786 

5274 
1548 

7345 
5456 

II. 

9876 
4894 

Addition. 

6721 
3834 

2925 
1673 

8007 
5002 

1439  1247  1020  1300  1172  1526  1000  1210 

1256  1072  1513  1040  1094  1017  1200  1030 

1073  1364  1327  1000  1207  1432  2030  1140 

1142  1289  1156  1708  1165  1157  1005  1327 

1381  1070  1208  1159  1082  1215  1234  1456 

1574  1528  1097  1023  1521  1000  1357  1579 

Addition  at  Sight, 

The  right-hand  figure  in  the  Sum  will  be  unchanged 
so  long  as  the  right-hand  figures  in  the  numbers  to  be 
added  remain  unchanged,  however  we  vary  the  Tens. 
Thus ;  3  and  5  are  8 ;  13  and  5  are  18 ;  13  and  15  are 
28.  Changing  the  Tens  in  the  numbers  to  be  added 
changes  the  Tens  in  the  Sum,  but  not  the  Ones, 

1  11      11      2      12      12      1      11      11      2     12     12 

2  2      12      2        2      12      3        3      13      3       3      13 


72  FIBST  LESSONS  IN 

LESSON  XL/X. 

^BZATIOjY    ^BTyTjEJBJSr    ^D^ITIOJV  ^jV!2> 
SU'BT^A  CTIOJV, 

Example  1.  What  is  the  Sum  of  347  and  456  ? 
Adding  as  in  other  cases  the  Sum  is 

803.  ADDITION. 

,  Example  2.    The  Sum  of  two        347)      Two 
numbers  is  803,  and  one  of  them  is        ^^ )  Numbers. 
456  ;  what  is  the  other  number  ?  803     Sum. 

Explanation^.  We  notice 
that  the  Minuend  is  like  the  subtraction. 

Sum  just  obtained  by  Addi-      Sum,      803  Minuend. 
tion ;  and  also  that  the  Sub-    ^One     l^^g  SuUrahend. 

trahend  is  one  of  the  two    Number.) 

numbers   just   added.     By 

subtracting  we  shall  obtain,  for  the  Difference,  the  other 

of  the  two  numbers  added. 

Since  we  cannot  take  6  Ones  from  3  Ones,  we  seek 
in  the  column  of  Tens  1  Ten  to  carry  to  the  right,  but 
find  none.  We  will  go  back  to  our  work  in  Addition 
and  find  the  reason  for  this. 

Adding  6  Ones  and  7  Ones,  we  had  for  the  Sum  13 
Ones ;  or  10  Ones  and  3  Ones.  We  wrote  the  3  Ones, 
and  carried  the  10  Ones,  as  1  Ten,  to  the  left.  Adding 
the  5  Tens,  4  Tens,  and  the  1  Ten  brought  from  the 
column  of  Ones,  we  had  10  Tens.  These  10  Tens  we 
carried  to  the  left,  as  1  Hundred,  and  wrote  no  Tens, 
One  of  these  Tens  came  from  the  column  of  Ones. 
Thus,  in  adding  456  and  347,  we  carried  10  Ones  not  only 
to  the  column  of  Tens,  but  afterwards,  with  the  9  Tens, 
to  the  column  of  Hundreds.  Now,  when  we  come  to 
Subtract  456  from  this  Sum,  803,  to  obtain  the  other 
number,  we  cannot  Subtract  until  after  seeking  our  10 


MENTAL  AND    WRITTEN  ARITHMETIC, 


73 


Ones,  and  also  our  9  Tens,  in  the  column  of  Hundreds, 
wliere  loe  left  them,  and  carrying  them  lack  to  the  right. 

We  take  1  of  the  8  Hundreds,  and,  carrying  it  to  the 
column  of  Tens,  separate  it  into  9  Tens  and  1  Ten ;  and 
leaving  the  9  Tens  in  the  column  of  Tens,/rom  which 
ive  formerly  carried  them,  we  then  carry  the  1  Ten  to 
the  column  of  Ones,  from  which 
it  had  been  carried,  and  write  it  as  solution. 

10  Ones.  Subtracting  6  Ones  from 
10  Ones  and  3  Ones,  5  Tens  from 
9  Tens,  and  4  Hundreds  from  7 
Hundreds,  the  Difference  is  347, 
the  other  number. 


7  9-10  J  Minuend 
$  0  3  j  re-arranged. 
4  5  6     Subtrahend, 

3  4  7    Difference, 


INFERENCES, 

I. — If  ^e  unite  two  numbers  by  Addition",  we  may 
disunite  them  by  Subtractiok. 

II. — If,  in  adding  two  Numbers,  we  carey  to  the 
LEFT,  in  Subtracting  either  ISTumber  from  the  Sum  to 
find  the  other,  we  must  carry  to  the  right  the  same 
Numbers  Avhich  we  carried  to  the  left  in  adding. 

Exercises  for  the  Slate  A]srD  Board. 


■  Addition: 

f  4526 
1375 

f  2746 
5273 

f  3254 
1647 

f  3127 
4375 

1. 

Subtraction  : 

5901 
1375 

2. 

3. 

8019 
5273 

4.. 

4901 
1647 

7502 
.  4375 

Addition: 

ri542 
3486 

r  6849 
2053 

'  1872 
4029 

r  3529 
4490 

3. 

Subtraction  : 

5028 
1543 

6. 

8902 
6849 

8.- 

5901 

,  1872 

8019 
.3529 

74  MRST  LESSONS  IN 


LESSON   L. 

By  turning  now  to  page  68,  you  will  see  that  tlie 
Thousand-group  of  blocks  is  in  the  form  of  a  cube; 
being  10  blocks  in  hight,  in  width,  and  in  length.  It 
contains  One  Thousand  blocks. 

This  same  mechanic  made  Alphabet  Blocks  in  large 
quantities,  and  used  to  pile  them  up  in  cubical  Thou- 
sand-groups. Then,  wrapping  a  strong  paper  around 
each  group,  he  tied  up  the  Thousand  blocks  firmly  in  a 
package. 

Having  a  yery  large  number  of  such  packages  on 
hand  one  day,  he  requested  his  little  son  and  one  of  the 
workmen  to  count  them,  with  the  blocks  previously 
counted  by  the  boy. 

They  counted  the  packages  in  Ten-groups;  then 
counted  the  Ten-groups  by  Tens,  to  make  Hundred- 
groups  ;  then  counted  these  Hundred-groups  by  Tens, 
to  make  Thousand-groups. 

They  found  that  there  were  just  as  many  pacTcages 
now  as  there  were  single  UocJcs  formerly  counted  by  the 
boy ;  as  shown  on  page  68.  They  were  piled  up  and 
arranged  in  the  same  manner.  At  the  right  was  1  sin- 
gle package ;  at  the  left  of  this  1  Ten-group ;  next  1 
Hundred-group;  and  last,  at  the  left,  1  Thousand- 
group. 

They  now  placed  their  single  package  at  the  left  of 
the  blocks  previously  counted  by  the  boy,  close  by  the 
side  of  his  Thousand-group  of  blocks,  which  they  tie(|^ 
up  as  one  package.  At  the  left  of  these  they  placed  their 
Ten-group  of  packages;  at  the  left  of  this  their  Hun- 
dred-group of  packages ;  and  last  their  Thousand-group 
of  packages. 


MENTAL  AND    WRITTEN  ARITHMETIC. 


75 


The  blocks  and  groups  then  stood  thus : 


Cube  op 
Packages. 


Packages. 


Blocks. 


P^ 

^ 

P. 

?  ^ 

p 

P 

, 

One   - 
sand-grc 
Package 

o 

cL 

_J! 

Q 

p^ 

One 
dred-gr 

One 
5n-grou 

Two 

ackages 

One 
dred-gr 

One 
m-grou 

One 
Block. 

Ah 

H 

Said  the  boy :  "  The  blocks  not  in  packages  stand  at 
the  right,  and  are  One  Hundred  and  Eleven.  We  will 
write  the  number  of  these  thus:  111.'' 

The  man  replied:  "Our  packages,  like  our  blocks, 
are  Cubes.  And  since  we  have  counted  and  arranged 
them  in  the  same  manner  and  order  as  we  did  the 
blocks,  we  will  also  tvrite  the  figures  showing  their  num- 
ber in  the  same  manner  and  order  as  we  did  those  show- 
ing the  number  of  single  blocks.  We  have  2  packages, 
1  Ten-group  of  packages,  and  1  Hundred-group  of  pack- 
ages; or.  One  Hundred  and  Twelve  packages.  Since 
these  stand  at  the  left  of  our  111  blocks,  we  will  write 
the  figures  112  at  the  left  of  111 ;  placing  a  mark  like  a 
comma  between  the  two  groups  of  figures;  thus,  112,111. 
Each  of  these  112  packages  contains  a  Thousand  blocks. 
Hence  the  figures  112,111  are  read:  One  Hundred  and 
Twelve  Thousand  One  Hundred  and  Eleven. 

"Since  we  have  1  very  large  cube,  containing  One 
Thousand  packages,  or  10  Hundred-groups  of  packages, 
which  stands  at  the  left  of  our  112  packages,  we  will 
write  for  this  a  figure  1,  placing  it  at  the  left  of  112, 
with  a  point  after  it;  thus,  1,112,111. 

"  This  very  large  cube  of  packages  is  named  a  MJitf 
lion.    A  Thousand  is  a  large  cube  containing  a  Thou- 
6 


76  FIRST  LESSONS  IN 

sand  Hocks,  A  Million  is  a  very  large  cube  containing 
a  Thousand  TJiousandsr 

If,  now,  we  had  more  Millions,  as,  for  instance,  Three 
Hundred  and  Sixty-five  Millions,  instead  of  One  Mil- 
lion, we  would  write  them  by  placing  the  figures  365 
at  the  left  of  112,111 ;  thus,  365,112,111.  We  see  that, 
commencing  at  the  right,  we  have  separated  the  figures 
of  this  number  into  groups  each  having  three  figures. 
Each  group  of  figures  is  named  a  Period.  The  point, 
made  like  a  comma,  separating  the  Periods  from  each 
other,  is  named  a  Period-point. 

The  manner  of  separating  a  number  having  nine 
figures  into  Periods,  and  naming  the  Periods  and  the 
Subdivisions  in  each,  is  shown  in  the  following 


TABZD 

'• 

THREE  PERIODS. 

Numbered, 

3d. 

2d. 

1st. 

Millions. 

Thousands. 

Ones. 

~> 

1 

^ 

x& 

{d 

^ 

^ 

x& 

C3 

o 

OQ 

Subdivided 

1 

1 

s 

m 

INTO  AND 

Named, 

g 

1 

O 

1 

O 

a;* 

ca 

S 

w 

H 

a 

m 

H 

H. 

w 

H 

o 

3 

6 

3  > 

,  1 

1 

2  , 

.  1 

J 

1 

All  the  Periods  are  built  up  in  the  same  manner,  each 
from  its  own  cube.  Since  the  cube  from  which  the 
right-hand  Period  is  built  up  is  One^  or  One  block,  this 
Period  is  named  the  Period  of  Ones.  The  middle 
Period,  having  a  Thousand  for  its  cube,  is  named  the 
Period  of  Hiousands.  For  a  like  reason  the  left- 
hand  Period  is  named  the  Period  of  Millions. 


MENTAL  AND    WRITTEN  ARITHMETIC. 


77 


LESSON   LI. 

JVOTATIOJV  AJ\r2)  J\rirM£:'RATIOJV. 


TABIjE, 

1  One                      is  written               1 

10  Ones 

are  1  Ten;                        written              10 

10  Tens 

"    1  Hundred;                     * 

100 

10  Hundreds 

"     1  Thousand; 

1,000 

10  Thousands 

"     1  Ten-thousand;       '      ' 

10,000 

10  Ten-thousands 

"     1  Hundred-thousand;    * 

100,000 

10  Hundred-thousands 

"    1  Million  ; 

1,000,000 

10  Millions 

"     1  Ten-million; 

10,000,000 

10  Ten-millions 

"     1  Hundred-million          ' 

'    100,000,000 

To  write  numbers  in  figures :  Commence  at  the  left 
and  ivfite  the  periods,  one  after  another,  in  the  same 
order  as  the  words  are  written. 

Write  171  JFigures  the  following  : 

L  Five  Thousand ;  Four  Hundred ;  Seven  Thousand 
and  Twenty;  Nine  Thousand  and  Seven;  Thirteen 
Thousand,  and  Eleven. 

2,  One  Hundred  and  Three  Thousand,  One  Hundred 
and  Seventeen ;  Three  Hundred  Thousand,  and  Twenty- 
five  ;  Five  Millions,  One  Hundred  and  Eleven  Thousand, 
and  Seventeen. 

To  read,  or  numerate,  a  number  written  in  figures : 

1st.  Beginning  at  the  right,  separate  the  nvmber  into 
periods  ; 

2d,  Beginning  at  the  right,  name  all  the  periods; 

3d.  Beginning  at  the  left,  read  the  periods  in  order, 
giving  the  name  of  every  period  except  that  at  the  right, 

JV^umerate  the  following  JV^iimbers  : 

137256984 ;       40576402 ;       170510001 ;       103520407 ; 

19030040 ;       21005000 ;       100301200 ;         31001501. 


T8 


FIRST  LESSONS  IN 


LESSON   L/L 

First  numerate  the  numbers  in  each  of  the  following 
Exercises,  and  then  find  their  Sum. 

EXEKCISES   FOR  THE   SlATE  AlJfD   BOARD. 


35467 

127508 

I. 
1234567 

10154125 

12536421 

12243 

105623 

2514360 

12253326 

12037058 

10516 

100510 

1602507 

15107908 

10250870 

27638 

209018 

1019015 

11015017 

20900500 

32507 

234126 

II. 
3509215 

27514310 

12141871 

10325 

576387 

5280317 

41310080 

27423612 

47018 

154218 

4052001 

60040085 

12751423 

53106 

627324 

6003200 

21005002 

20205602 

61007 

542987 

1100005 

13241576 

10030050 

27589 

173238 

7020050 

11376529 

10400800 

Notation,  Nutneration,  and  A.ddltion, 

In  each  of  the  following  Examples,  first  write  the 
numbers,  then  numerate  them,  ^nd  finally  add  them. 

Exercises  for  the  Slate  a:n^d  Board. 

I.  Twenty  thousand,  One  Hundred  and  Five;  Thir- 
teen Thousand,  and  Fifteen ;  Seventeen  Thousand,  and 
Nine. 

II.  One  Hundred  and  Twenty-five  Thousand,  Three 
Hundred  and  Eleven ;  Three  Hundred  and  Seven 
Thousand,  Five  Hundred  and  Four;  Five  Hundred  and 
Eleven  Thousand,  and  Fifteen. 

III.  Three  Million,  Five  Hundred  and  Twenty-five 
Thousand,  One  Hundred  and  Twenty-Seven ;  Five  Mil- 
lion, Three  Hundred  and  Seven  Thousand,  Seven  Hun- 
dred and  Eight;  Nine  Million,  Five  Thousand,  and  Six. 


MENTAL  AND    WRITTEN  ARITHMETIC,  79 


LESSON  Llll. 

SZrSTlici.CTIOJV. 

867,542 
573,814 

5,473,207 
1,632,154 

I. 

63,521,365 
21,507,193 

750,319,476 
473,207,528 

953,541 
684,703 

2,507,318 
1,498,173 

II. 
58,390,504 
28,279,371 

972,548,765 
481,395,976 

Addition  and  Subtraction. 

Writtei^  Exercises. 

1,  Three  grain  dealers  purchased  wheat  as  follows: 
A  purchased  59,487  bushels  ;  B,  47,376  ;  and  C,  39,894. 

(a.)  How  many  bushels  did  the  three  men  purchase  ? 
{b.)  How  many  bushels  did  A  and  B  together  pur- 
chase more  than  C  ? 

2,  A  father  divided  his  property  among  his  children 
as  follows:  He  gave  to  Charles,17,510  dollars;  to  Henry, 
21,437  dollars ;  to  William,  25,196  dollars ;  to  Amelia, 
13,087  dollars ;  to  Sarah,  15,193  dollars ;  and  to  Susan, 
11,981  dollars. 

{a.)  How  many  dollars  did  he  give  to  his  three  sons  ? 
(J.)  How  many  dollars  to  his  three  daughters? 
{c.)  How  many  dollars,  in  all,  to  his  children  ? 
{d.)  How  many  more  dollars  were  given  to  his  sons 
than  to  his  daughters  ? 

3,  In  the  year  1850,  the  population  of  the  city  of  New 
York  was  515,547;  Boston,  136,881;  Philadelphia, 
340,045, 

'    («.)  What  was  the  total  population  of  the  three  cities? 
(J.)  How  much  did  the  population  of  New  York  ex- 
ceed that  of  both  Philadelphia  and  Boston  ? 


80  FIRST  LESSONS  IN 

LESSON   LIV. 

JVUMB^ATIOJV  ciJY^   ^^^ITIOJSr. 

First  read,  or  numerate,  the  numbers,  and  then  find 
their  Sum,  in  each  of  the  following 

Exercises  for  the  Slate  Aiq"D  Board. 


33,507 

234,126 

3,509,215 

27,514,310 

12,418,712 

10,325 

576,387 

5,280,317 

41,310,080 

27,236,129 

47,018 

154,218 

4,052,001 

60,040,085 

12,514,237 

53,106 

627,324 

6,003,200 

21,005,002 

20,056,025 

61,007 

542,987 

1,100,005 

13,241,576 

10,300,500 

27,589 

173,238 

7,020,050 

11,376,529 

10,008,006 

Notation  and  Addition, 

In  each  of  the  following  Examples,  first  write  the 
numbers,  then  numerate  them,  and  finally  find  the  Sum. 

Examples  for  the  Slate  ai^t>  Board. 

I.  Twenty  Thousand,  One  Hundred  and  five ;  Thir-. 
teen  Thousand,  and  fifteen ;  Seventeen  Thousand,  and 
Nine. 

II.  One  Hundred  and  Twenty-five  Thousand,  Three 
Hundred  and  Eleven ;  Three  Hundred  and  Seven  Thou- 
sand, Five  Hundred  and  Four;  Five  Hundred  and 
Eleven  Thousand,  and  Fifteen. 

III.  Three  Million,  Five  Hundred  and  Twenty-four 
Thousand,  One  Hundred  and  Twenty-seven ;  Five  Mil- 
lion, Three  Hundred  and  Seven  Thousand,  Seven  Hun- 
dred and  Eight ;  Nine  Million,  Five  Thousand,  and  Six. 

IV.  Two  Hundred  and  Seventy-six  Million,  Three 
Hundred  and  Seven  Thousand,  One  Hundred  and  Nine- 
teen ;  One  Hundred  and  Two  Million,  Forty-five  Thou- 
sand, and  Twelve;  Five  Hundred  Million,  Eight  Thou- 
sand, and  Nine. 


MENTAL  AND    WRITTEN  ARITHMETIC. 


81 


/■    /    r   ^ 

k 

fc 

t    h^  'fc.  \ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

■™ 

^ 

^ 

^ 

_ 

^ 

4 
4 
8 


5 
]0 


6 
_6 
12 


7  8 
_7  _8 
14    16 


9 
_9 
18 


f    r    ^   ^  ^_  \ 

fc 

^      1 

^  '^  \ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2 

4 

6 

8 

iO 

12 

14 

16 

j^ 

'      LESSON  LV. 

These  first 
two  horizontal 
rojN^s  of  cubes 
were  begun  at 
the  left  with 
one  cube  in 
each,  and 
lengthened  by 
adding  another 
cube  to  each 
from  time  to 
time,  till  each 
row  had  9 
cubes.  The 
cubes  in  the 
upper  row  are  numbered. 

Below  the  figure  1  written  on  the  first  cube  in  the 
upper  row  we  have  written  two  figure  l^s,  and  have 
written  their  Sum  below,  to  show  how  many  cubes  there 
were  in  both  rows  when  there  was  1  cube  in  each  row. 

Under  the  2  written  in  the  upper  row  we  have  written 
two  figure  2's,  and  below  them  their  Sum,  to  show  that 
there  were  4  cubes  in  both  rows  when  there  were  2  cubes 
in  each  row. 

In  the  same  manner  we  have  found  the  number  of 
cubes  in  the  2  rows  at  each  point  from  their  commence- 
ment till  their  completion. 

In  the  second  cut  we  have  written  these  Sums  on  the 
cubes  in  the  lower  row.  We  see  that  when  there  were 
3  cubes  in  each  row  there  were  6  cubes  in  both  rows ; 
when  there  were  4  cubes  in  each  row  there  were  8  in 
both  rows. 


82  FIRST  LESSONS  IN 

When  the  numbers  added  to  form  a  Sum  are  equals 
Tve  name  the  Addition  Gn^aded  Addition^  since 
the  Sum  is  composed  of  equal  or  graded  parts. 

Since  we  found  these  Sums  by  adding  two  I's,  two 
2's,  two  3's,  &c.,  it  is  plain  that  the  Sums  show  how 
many  2  times  1,  2  times  2,  2  times  3,  &c.,  are.  Hence, 
our 

GMADED  ADDITION  TABLE. 

2  Times 

1  are  2,  4  are    8,  7  are  14, 

2  are  4,  5  are  10,  8  are  16, 

3  are  6,  6  are  12,  9  are  18. 

When  we  subtract  one  number  from  another  not  once 
only,  but  as  many  times  as  possible,  we  name  this  Sub- 
traction Graded  Subtraction. 

We  may  also  consider  these  cubes  as  piled  up  in  ver- 
tical columns,  each  having  2  cubes. 

From  the  first  column,  or  2  cubes  at  the  left,  we  can 
Subtract  2  cubes  once.  From  the  first  2  columns,  or  4 
cubes  at  the  left,  we  can  Subtract  2  cubes  2  times ;  from 
6  cubes  3  times;  from  8  cubes  4  times;  and  so  on. 
Hence  we  can  make  the  following,  or  First  Form  of 

G  BAD  ED  STTBTBACTION  TABLE, 

3  can  be  Subtracted  from 

2    1  Time,  8    4  Times,  14  7  Times, 

4    2  Times,  10    5  Times,  16  8  Times, 

6    3  Times,  12     6  Times,  18  9  Times. 

Since  2  cubes  are  contained  in  any  number  of  cubes 
as  many  times  as  they  can  be  Subtracted  from  that  num- 
ber of  cubes,  it  is  evident  that  we  may  have  the  follow- 
ing, more  common,  or  Second  Form  of 


MENTAL  AND    WRITTEN  ARITHMETIC. 


83 


GRADED   SUBTRACTION  TABLE. 

3  are  contained  in 

2    1  Time,  8    4  Times,  14  7  Times, 

4    2  Times,  10    5  Times,  16  8  Times, 

6    3  Times,  12     6  Times,  18  9  Times. 

We  can  read  these  Tables  from  the  cubes  as  a 

GRADED   ADDITION  AND   SUBTRACTION  TABLE. 


f 

r 

f   ~^ 

^ 

fc 

it   \ 

r  f^  ^ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

ki 

4 

6 

8 

10 

12 

14 

16 

18 

To  read  this  as  a  Graded  Addition  Table,  we  com- 
mence with  the  2  at  the  bottom,  at  the  left;  then 
take  a  number  in  the  upper  row,  as  3  ;  and  lastly  take, 
in  the  lower  row,  the  number  below  that  taken  in  the 
upper  row,  which  in  this  case  would  be  6 ;  saying :  2 
times  S  are  6,  In  like  manner  we  proceed,  saying :  2 
times  If.  are  8  ;  2  times  5  are  10  ;  and  so  on. 

To  read  it  as  a  Graded  Subtraction  Table,  we  com- 
mence with  the  2  at  the  left,  as  before,  but  take  the  two 
other  numbers  in  a  contrary  order ;  reading  first  the  2 
at  the  left ;  then  a  number  in  the  lower  row,  as  6  ;  and 
lastly  the  number  3  above  this ;  saying :  2  are  in  6,  S 
times.  Thus  we  go  on,  saying :  2  are  in  8,  4  times  j  2 
are  in  10,  5  times  ;  and  so  on. 

This  form  of  the  Tables,  as  represented  ly  the  cubes, 
is  the  more  convenient. 

These  Tables  should  be  perfectly  learned,  and  often 
recited. 


84  FIRST  LESSONS  IN 

LESSON    LVI. 

Writtei^  Mejsttal  Exercises. 

L  Two  horses  make  1  span.  How  many  horses  are 
there  in  2  spans  ?    2  times  2  horses  are  how  many  ? 

2,  One  wagon  has  4  wheels.  How  many  wheels  have 
2  wagons  ?    2  times  4  wheels  are  how  many  wheels  ? 

<^.  Fanny  found  2  birds'  nests,  each  having  6  eggs. 
How  many  eggs  were  there  in  all  ?  2  times  6  eggs  are 
how  many  eggs  ? 

Jf.  In  each  of  2  windows  there  are  8  panes  of  glass. 
How  many  panes  are  there  in  both  windows  ?  2  times 
8  panes  are  how  many  ? 

5,  Willie  had  two  rose-bushes  with  3  roses  on  each. 
How  many  roses  had  he  on  both  bushes  ?  2  times  3 
roses  are  how  many  roses  ? 

6,  Counting  my  thumbs  as  fingers,  I  have  5  fingers 
on  each  hand.  How  many  fingers  have  I  on  both 
hands  ?    2  times  5  fingers  are  how  many  ? 

7,  Frank  had  7  peaches,  and  Mary  also  had  7.  How 
many  had  both  ?    2  times  7  peaches  are  how  many  ? 

8,  Two  hens  have  each  9  chickens.  How  many 
chickens  have  both  hens  ?    2  times  9  are  how  many  ? 

How  many  times  can  we  Subtract 

2    wagon-wheels    from    8  2  horses  from  4  horses  ? 

wheels  ?  2  fingers  from  10  fingers  ? 

2  birds'  eggs  from  12  eggs  ?  2  roses  from  6  roses  ? 

2  chickens  from  18  chick-  2  window-panes  from   16 

ens  ?  panes  ? 

2  peaches  from  14  peaches  ? 

9,  How  many  cents,  in  all,  are  2  times  4  cents  anq 
2  times  3  cents  ? 


MENTAL  AND    WRITTEN  ARITHMETIC, 


85 


f     t     k-    r- 

L. 

fc 

^      ! 

fe--  '^.  \ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2 

4 

6 

8 

10 

12 

14 

16 

18 

_  3 

6 

9 

12 

15 

18 

21 

24 

27 

4 

4 

_4 

12 


5 

5 

_5 

15 


6 
'6 
_6 
18 


7  8  9 

7  8  9 

_7  _8  _9 

21  24  27 


LESSON   LVIL 

In  this  cut, 
the  2  horizon- 
tal rows  of 
cubes  shown 
in  the  last  cut 
are  seen  placed 
over  another 
row  of  cubes. 
Below  these 
cubes  numbers 
are  written  in 
vertical  col- 
umns. The  three  4's,  and  the  12  written  below  them, 
which  is  their  Sum,  show  that  when  there  were  4  cubes 
in  each  of  the  3  horizontal  rows  there  were  12  cubes  in 
the  3  rows.  So  of  the  other  columns  of  figures  and 
their  Sums.  These  Sums  are  written  on  the  cubes  in 
the  lower  row. 

The  new  part  added  to  the  Table  is  read  thus  as  a 
part  of  the 

GBADEJ)    ADDITION    TABLE, 

3  Times 

4  are  12, 

5  are  15, 

6  are  18, 
It  is  read  thus  as  a  part  of  the 

*GItADED    SUBTRACTION    TABLE, 

3  can  be  Subtracted  from 

1  Time,  12    4  Times,  21 

2  Times,  15     5  Times,  24 

3  Times,  18     6  Times,  27 
Eepeat  the  entire  Table ;  first  as  a  Graded  Addition 

Table,  and  then  as  a  Graded  Subtraction  Table. 


1  are  3, 

2  are  6, 

3  are  9, 


3 

6 
9 


7  are  21, 

8  are  24, 

9  axe  27. 


7  Times, 

8  Times, 

9  Times. 


8G  FIRST  LESSONS  IN 

LESSON   LVIII. 

Writtei^  Mei^tal  Exercises. 

1,  Charles  found  3  birds'  nests,  each  having  4  eggs. 
How  many  eggs  were  there  in  the  3  nests  ? 

2,  Emma  attended  school  5  days  each  week  for  3 
weeks.    How  many  days  did  she  attend  in  3  weeks  ? 

3,  In  a  school  are  3  classes  in  reading,  with  9  pupils 
in  each  class.     How  many  pupils  in  the  3  classes? 

-4.  Each  of  3  boys  has  8  cents.  How  many  cents  have 
the  3  boys  ?     3  times  8  are  how  many  ? 

5,  Three  boys  went  fishing,  and  each  caught  7  fish. 
How  many  fish  did  the  3  boys  catch  ? 

6,  A  board  fence  was  6  boards  high.  How  many 
boards  were  there  in  3  lengths  of  the  fence  ? 

7.  A  father  had  12  apples,  and  gave  them  to  his  sons, 
giving  each  boy  3  apples.  How  many  sons  were  there  ? 
3  apples  are  in  12  apples  how  many  times  ? 

8,  A  company  of  girls  picked  18  quarts  of  berries, 
each  girl  picking  3  quarts.  How  many  girls  were  there  ? 
3  are  in  18  how  many  times  ? 

9,  Florence  had  15  roses  in  her  flower-garden,  and 
there  were  3  roses  on  each  bush.  How  many  rose-bushes 
were  there  ?    3  are  in  15  how  many  times  ? 

10.  A  beggar  received  24  cents  from  a  company  of 
boys,  each  boy  giving  him  3  cents.  How  many  boys 
were  there  ?     3  are  in  24  how  many  times  ?  . 

11.  Willie  had  27  cents  in  3-cent  pieces.  How  many 
3-cent  pieces  had  he  ?     3  are  in  27  how  many  times? 

12.  A  mother  had  21  flowers,  and  made  bouquets  of 
them  for  her  children,  putting  3  flowers  in  each  bou- 
quet. How  many  children  had  she  ?  3  are  in  21  how 
many  times  ? 


MENTAL  AND    WRITTEN  ARITHMETIC, 


87 


LESSON   LIX. 

MUZ  TITZICA  TIOjV. 

Example.  A  tailor  cut  from  a  roll  of  clotli  enoiigli 
for  7  pairs  of  boys'  pants,  using  2  yards  for  each  pair. 
How  many  yards  did  he  use  ? 

ExPLAN"ATiON".  He  first  measured  2  yards  at  one  end 
of  the  roll,  and  then  hj  folding  the  cloth  first  one  way 
and  then  the  other,  until  he  had  7  thicknesses,  he  thus 
measured  7  times  2  yards. 

In  thus  obtaining  7  times  2  yards,  he  folded  the  2 
yards  many  times.  Taking  2  yards  7  times,  and  finding 
how  many  yards  we  have,  is  sometimes  named  Jfulti-^ 
plication.  Multiplication  me'dixs  folding  many  times, 
or  taking  many  times.    We  have  taken  2  yards  7  times. 

We  name  the  ^  yards  the  Multiplicand.  Multi- 
plicand  means  something  to  he  folded  or  tahen  many 
times. 

We  name  7  the  Multiplier.  The  Multiplier  tells 
how  many  times  the  Multiplicand  is  to  be  taken. 


88 


FIRST  LESSONS  IN 


There  are  two  Solutions  for  this  Exam- 
ple. First:  We  write  a  figure  2  seven 
times,  and,  adding  the  2's,  write  the  Sum, 
14,  below. .  This  method  of  performing 
the  work  we  name  Graded  Addition. 
Second :  We  write  the  2  once  only,  and 
then  writing  a  figure  7  under  it,  to  show 
how  many  times  the  2  is  to  he  taken,  draw 
a  line  under  the  7,  and  saying :  "  7  times 
2  are  H,''  write  14  below  the  line. 
This  method  of  performing  the  work 
we  name  Multiplication,    The  result 


First  Solution. 
By  Addition. 
2' 

^      Equal 

2  1      ^"^ 
2  I  Graded 

2      Parts. 

_2^ 

14  Sum. 


is  14  in  both  cases ;  but  the  14  ob- 


Second  Solution. 
By  Multiplication. 

2  Multiplicand* 
7  Multiplier. 
14  Product. 


tained  by  the  first  Solution  is  named 
the  Sum,  and  that  obtained  by  the 
second  Solution  is  named  the  Product.  Product 
means  something  produced.  H  is  produced  by  multi- 
plying 2  by  7. 


EXEKCISES  POR  THE   SlATE 

AND  Board. 

Graded  Addition 

, 

3  Equal 

or  Graded  ■ 

Parts. 

'4      20 

4      20 

.4      20 

24 
24 
24 

100 
100 
100 

124      3,000 
124      3,000 
124      3,000 

3,124 
3,124 
3,124 

- 

Multiplication, 

Multiplicands:  4 
Multipliers :      3 

20 
_3 

X. 

24      100 
3          3 

124    3.000 
3           3 

3,124 
3 

51,323 
3 

93,423 

2 

II. 

83,213 
3 

50,706 

2 

90,708 
3 

MENTAL  AND    WRITTEN  ARITHMETIC, 


89 


FIRST  SOLUTION. 

By  Addition. 

5  +  5  +  5  =  15. 

SECOND  SOLUTION. 

_  By  Multiplication. 

5  X  3  =  15 


LESSON   LX. 

Example.  How  many  are  3  times  5  ? 

ExPLAN^ATiON^.  In  the  first  Solution  we  write  5  three 
times,  and,  adding,  write  the  Sum, 
15 ;  thus,  5  +  5  +  5  =  15.  In  the 
second  Solution  we  write  5  once 
only  ;  and,  since  it  is  to  be  taken  S 
times  and  added,  we  write  3  at  the 
right  of  5,  and,  turning  the  Sign 
Plus  thus,  X ,  place  it  hetiveen  the 
5  and  3.  The  whole  stands  thus : 
5  X  3  =  15 ;  and  is  read :  5  mul- 
tiplied by  3  equal  15.  When  the  Sign  Plus  is  thus 
turned  and  used  it  does  not  show  that  5  and  3  are  to  be 
added  together,  but  shows  that  5  are  to  be  taken  3  times 
and  added,  to  find  the  Sum;  or, 
which  means  the  same,  it  shows 
that  5  are  to  be  multiplied  by  3. 
For  this  reason,  when  the  Sign  Plus 
is  thus  turned  and  used  we  name 
it  the  Sign  of  Multiplica- 
tion. The  Sign  of  Addition  is  a 
Vertical  Cross.  The  Sign  of  Mul- 
tiplication is  an  Inclined  Cross, 

We  can  write 

6  multiplied  by  3  equal  18  ;  or  6  x 
9  multiplied  by  3  equal  27 ;  or  9  x 

Write,  learn,  and  recite,  the  following 
ta:bJjE, 


3  =  18; 

3  =  27. 


X  2  =  0 

x  2  =  2 

X  2  =  4 

X  2  =  6 

X  2  =  8 


2  =  10 
2  =  12 
2  =  14 
2  =:  16 
2  =  18 


3  =  0 
3  =  3 
3  =  6 
3  =  9 
3  =  12 


5  X 

6  X 

7  X 

8  X 

9  X 


=  15 
=  18 
=  21 
=  24 


3  =  27 


90  FIRST  LESSONS  IN 

LESSON   LX/. 

Example  A.  Multiply  879  by  3. 

ExPLAi^ATiojsr.   First ;  we  multiply  9  Ones  by  3,  and 
write  the  Product,  27  Ones.    We  name  this  a  Partial 
Product^   because    it   forms 
only  a  part  of  the  true  Pro-  solution. 

duct.     Second;  we  multiply  7         879    Multiplicand, 

Tens  by  3,  and,  having  21  Tens,      . §    Multiplier. 

or  2  Hundreds  and  1  Ten,  for  27  1     p    /  •  ? 

our    second    Partial    Product,         21    [p^^^J^^^^^ 
write  1  Ten  in  the  column  of        ^^     ^ 
Tens,  and  2  Hundreds  at  the      2,637    Product, 
left,  in  the  column  of  Hundreds. 

Third ;  we  multiply  8  Hundreds  by  3,  and,  having  24 
Hundreds,  or  2  Thousands  and  4  Hundreds  for  our 
third  Partial  Product,  write  4  Hundreds  in  the  column 
of  Hundreds,  and  2  Thousands  at  the  left.  Now  we 
have  taken  the  8  Hundreds  3  times,  the  7  Tens  3  times, 
and  the  9  Ones  3  times.  If  we  add  these  three  Partial 
Products  we  shall  have  3  times  879  for  the  final  Product. 

First,  we  bring  down  the  7  Ones  into  the  Product. 
Next,  we  add  1  Ten  and  2  Tens,  and  write  their  Sum,  3 
Tens,  in  the  Product.  Adding  4  Hundreds  and  2  Hun- 
dreds, we  write  their  Sum,  6  Hundreds,  in  the  Product. 
Finally,  we  write  2  Thousands  at  the  left  of  6  Hundreds, 
and  have  2,637  for  our  final  Product. 

EXEKCISES  FOR  THE   SlATE  AKD  BoARD. 


57,489 
3 

95,847 
3 

I. 

48,796 
3 

II. 

85,798 
3 

79,846 
3 

64,587 
3 

84,758 
3 

89,798 
3 

94,876 
3 

58,764 
3 

MENTAL  AND  WRITTEN  ARITHMETIC. 


91 


LESSON   LXIL 

Example  B.  Multiply  52,819  by  3. 

ExPLAi5"ATioi^.  The  work  in  this  Solution  is  placed 
in  nearly  the  same  order  as 
that  in  the  Solution  of  Ex- 
ample A,  The  second  and 
third  Partial  Products  are, 
however,  placed  in  the  same 
horizontal  line,  since  they  do 
not  interfere  with  each  other; 
so  also  the  fourth  and  fifth. 


52,819 
3 

271 
24-3 
15-6 


Multiplicand. 
Multiplier. 

Partial 
Products. 


158,457     Product. 

Exercises  for  the  Slate  akd  Board. 

I. 

584,319        769,815        715,394        428,173        936,284 


152,873 
3 


231,312 
3 


II. 
987,546 
3 


514,271 
3 


Example  C.  Multiply  50,608  by  3. 

ExPLAi^ATioi^.  Since  no  0  in  the  Multi- 
plicand gives  rise  to  any  Partial  Product, 
we  write  0  in  place  of  a  Partial  Product,  in 
such  cases. 


859,897 
3 

SOLUTION. 

50,608 
3 

24 
18-0 
15-0 

151,824 


Exercises  for  the  Slate  ai^d  Board. 

I. 

20,708    69,008    304,500    50,980    100,709 


2 


2 


2 


2 


20,030 
3 


58,090 
3 


II. 
600,708 
3 


12,500 
3 


907,008 
3 


92 


FIRST  LESSONS  IN 


4  Multiplicand. 
3  Multiplier. 

12  Product. 


LESSON   LXIII. 

These  12  apples  are  arranged  in  two  sets  of  rows ;  one 
set  running  lengthwise  of  the  table,  and  the  other 
crosswise. 

FiKST :  If  we  consider  the  rows  pour  m  a  row. 

as  running  lengthwise,  there  are  3 
rows.  In  each  row  are  4  apples. 
In  3  rows  there  are  3  times  4  ap- 
ples; which  are  12  apples.  The 
Multiplicand  is  4,  the  number  of  apples  in  each  row ; 
the  Multiplier  is  3,  the  number  of  rows ;  and  the  Pro- 
duct is  12,  the  number  of  apples  on  the  table. 

Second  :  If  we  consider  the  rows 
as  running  crosswise,  there  are  4 
rows.  In  each  row  are  3  apples, 
giving  3  for  the  Multiplicand.  In 
4  rows  there  are  4  times  3  npples ; 
which  are  12  apples ;  giving  4  for  a 
Multiplier,  and  12  for  the  Product. 

We  notice  that  the  two  numbers  multiplied  together 
are  the  same  in  both  cases ;  and  also  the  Products ;  but 


THREE  IN  A  ROW. 

3  Multiplicand. 

4  Multiplier. 

12  Product. 


MENTAL  AND    WRITTEN  ARITHMETIC,  93 

the  Multiplicand  and  Multiplier  of  the  first  change 
places  in  the  second. 

.  In  each  case  the  Multiplicmid  shows  the  number  of 
apples  in  a  row,  the  Multiplier  the  number  of  roivs,  and 
the  Product  the  number  of  apples  arranged, 

Inferen'CE.  I7i  all  cases  where  the  Product  represents 
MATERIAL  THiJS'GS,  as  apples,  peachcs,  oranges,  1st,  these 
things  can  be  so  arranged  as  to  stand  in  two  sets  of 
roios  ;  2d,  the  number  of  thiiigs  in  one  row  of  either  set 
mag  be  tahen  as  the  Multiplicand,  and  the  number  of 
things  in  one  row  of  the  other  set  as  Multiplier ;  and, 
3d,  the  number  of  things  arranged  will  be  the  Product, 

Example.  A  lady  gave  4  apples  to  each  of  3  boys, 
and  3  apples  to  each  of  4  girls ;  how  many  apples  did 
she  give  to  the  3  boys,  and  how  many  to  the  4  girls  ? 

Explain ATioN.  The  arrangement  of  apples  in  the 
foregoing  cut  will  give  us  both  answers  to  this  Example. 
*  In  the  first  case  there  are  4  apples  in  each  row,  and  3 
rows ;  or  one  row  for  each  boy.  4  x  3  =  12.  Hence 
12  apples  were  given  to  the  3  boys. 

In  the  second  case  there  are  3  apples  in  each  row,  and 
4  rows ;  one  row  for  each  girl.  3  x  4  =  12.  Hence 
12  apples  were  given  to  the  4  girls. 

3  and  4  multiplied  together  malce  ov  produce  12.  For 
this  reason  3  and  4  are  named  the  Factors  of  12. 
Factor  means  Maker  or  Producer,  12  is  the  thing  made, 
OY  produced,  and  hence  is  named  the  Product*  Pro- 
duct means  something  made  or  produced. 

From  the  foregoing  Explanations  we  infer  this 

FRINCIPIjE    in    3rULTirZICATION, 

If  two  Numbers  are  to  be  midtiplied  together  either 
may  be  used  as  the  Multiplicand,  and  the  other  as  the 
Multiplier. 


94 


FIRST  LESSOJSS  IN 


LESSON   LXIV. 

By  the  use  of  the  Principle  stated  in  the  preceding 
Lesson,  we  can  change  the  Multiplication  Tables  already 
learned  into  Tables  haying  1,  2  and  3  as  Multiplicands, 
and  1,  2,  3,  4,  5,  6,  7,  8  and  9,  as  Multipliers ;  thus : 


FinST  TABZE. 

SECOND   TABZE, 

3  Times  1  are    2, 

and  1  Time  3  is      2 ; 

2  Times  2  are    4, 

and  2  Times  2  are    4 ; 

2  Times  3  are    6, 

and  3  Times  2  are    6 ; 

2  Times  4  are    8, 

and  4  Times  2  are    8 ; 

2  Times  5  are  10, 

and  5  Times  2  are  10 ; 

2  Times  6  are  12, 

and  6  Times  2  are  12 ; 

2  Times  7  are  14, 

and  7  Times  2  are  14 ; 

2  Times  8  are  16, 

and  8  Times  2  are  16 ; 

2  Times  9  are  18, 

and  9  Times  2  are  18. 

3  Times  1  are    3,' 

and  1  Time  3  is      3  ; 

- 

3  Times  2  are    6, 

and  2  Times  3  are    6 ; 

3  Times  3  are    9, 

and  3  Times  3  are    9 ; 

3  Times  4  are  12, 

and  4  Times  3  are  12 ; 

3  Times  5  are  15, 

and  5  Times  3  are  15 ; 

3  Times  6  are  18, 

and  6  Times  3  are  18 ; 

3  Times  7  are  21, 

and  7  Times  3  are  21 ; 

3  Times  8  are  24, 

and  8  Times  3  are  24 ; 

3  Times  9  are  27, 

and  9  Times  3  are  27. 

Learn,  and  often  recite,  the  above  Tables. 

Mektal  Exercises. 

5 

X 

2  =:  ?        6x2  = 

?        7x2=?       2x6 

=  ? 

2 

X 

8  =  ?        8x2  = 

?        2x8=?        7x3 

=  ? 

3 

X 

4  --=  ?        3x6  = 

?        3x8=?        8x3 

=  ? 

2 

X 

5  =  ?       2x7  = 

?        2x9=?        9x2 

=  ? 

3 

X 

9  =  ?       3x5  = 

?       9x3=?       3x7 

=  ? 

MENTAL  AND    WRITTEN  ARITHMETIC. 


95 


LESSON   LXV. 

Exercises  for  the  Slate  aij^d  Board. 
I. 


2,132 
4 


30,201 
4 


3,213 
>   5 

1,323 
6 

II. 

2,131 

7 

3,231 

8 

2,333 
9 

21,032 
5 

23,103 
6 

31,232 

7 

13,233 

8 

33,232 
9 

Multiply  31,213,210  by  4 
Multiply  23,131,032  by  4 
Multiply  12,302,313  by  4 
Multiply  30,131,231  by  4 
Multiply  21,312,103  by  4 


;  by  5 

;  by  6; 

;  by  5 

;  by  6; 

;  by  5 

.  by  6; 

;  by  5 

by  6; 

;  by  5. 

byO; 

by  7; 
by  7; 
by  7; 
by  7; 
by  7; 


by  8. 
by  8. 
by  8. 
by  9. 
by  9. 


LESSON   LXVI. 

Exercises  for  the  Slate  akd  Board. 

Multiply  31,020,230  by  3 ;    by  4 ;    by  5 ;  by  6 ;  by  7. 

Multiply  12,130,302  by  3 ;    by  4 ;    by  5 ;  by  6 ;  by  7. 

Multiply  31,213,023  by  3 ;    by  4 ;    by  5 ;  by  6 ;  by  7. 

Multiply  21,302,013  by  3 ;    by  4 ;    by  5 ;  by  6 ;  by  8. 

Multiply  30,130,231  by  3 ;    by  4;    by  5;  by  6;  by  8. 


Multiplication  at  Sight, 


4  X 

2  X 

8  X 

2  X 

7  X 


X  2 

X  4 

X  8 

X  6 

X  2 


.3 
5 

8 
2 
9 


96 


MBST  LESSONS  IN 


LESSON   LXVII. 

Example. — Harry's  mother  gave  12  apples  to  her 
children,  giving  4  apples  to  each.  How  many  children 
had  she  ? 

Explaintatiok.  —  1st: 
Subtracting  4  apples  from 
12  apples,  we  have  8  ap- 
ples left.  These  4  apples 
are  arranged  in  the  1st  row 
in  the  picture.  2d:  Sub- 
tracting 4  apples  from  8 
apples,  we  have  4  apples 
left.  The  4  apples  last 
subtracted  are  arranged  in 

the  2d  roio.  3d :  Subtracting  4  apples  from  4  apples,  we 
find  none  remaining.  These  4  apples  are  arranged  in 
the  3d  row. 


FIRST   SOLUTION. 

By  Graded  Subtraction. 

12  ajjples.    Minuend, 


1st  SuUraliend. 


2d 


3d 
No  apples  left. 


MENTAL  AND    WRITTEN  ARITHMETIC.  97 

In  this  manner  we  divide  12  apples  into  3  groups, 
each  having  J^.  ap2)les.  Since  ^  apples  were  given  to 
each  child,  the  number  of  groups  and  the  nurnber  of 
children  must  have  been  the  same.  Hence  there  were 
3  children. 

Testing  the  Result, 

If  12  apples  can  be  divided  into  3  tbst. 

groups,  each  having  4  apples,  then  3     ^^  Graded  AddUim, 
times  4  apples  must  equal  12  apples.       -z^^     ^  Apples. 
Therefore,  we  write  three  4's  and  add     -  q^      a        « 
them.     Finding  the  Sum  to  be  12,  we  — 

conclude  that  there  must  have  been  3     ^^^^  ^^ 
groups,  with  4  apples  in  each.     Hence  there  were  3 
children. 

Secokd.  —  We  can  perform  our  work  by  another 
method,  and  find  the  same  result.  We 
write  4  apples,  and,  drawing  a  line      second  solution. 
below,  write  12  apples  below  this  line.        ^  Groups. 
We  then  ask :  Hotv  many  times  can  4      _Z  Apples. 
apples  be  subtracted  from  12  apples  ?       12      " 
or,  Hoio  many  times  are  4  apples  con- 
tained in  12  apples  ?     Finding  that  4  are  in  12  3  times, 
we  write  3  above  the  4  apples,  to  show  how  many  groups 
there  are.    Since  12  apples  can  be  divided  into  3  groups, 
each  having  4  apples,  Harry's  mother  must  have  had  3 
children. 

Testing  the  Result, 

If  there  are  in  12  apples  3  groups,  each  having  4 
apples,  then  there  are  in  all  3  times  4  apples;  which 
are  12  apples.  Since  3  and  4,  in  the  Solution,  stand 
over  12  in  the  same  manner  as  the  Multiplicand  and 
Multiplier  stand  over  their  Product,  we  will  multiply 
them  together. 


98  FIRST  LESSONS  IN 

Using  3  as  Multiplier,  the  Product  is  12  apples. 
Hence,  in  12  apples  there  are  3  groups,  each  having  4 
apples.     Therefore,  3  children  received  12  apples. 

The  method  which  we  have  just  used,  in  the  Second 
Solution,  is  named  Division.  12  is  named  the 
Dividend,  4  the  Divisor,  and  3  the  Quotient. 

L  DivisiOi?"  means  dividing.  We  have  divided  12 
apples. 

2,  Dividend  means  something  to  le  divided.  12  is 
the  number  to  he  divided. 

S.  Divisor  means  divider.  We  have  used  4  as  a 
divider  of  12. 

Jf.  QuoTiEi^T  means  how  man^  times.  3  shows  how 
many  times  4  are  contained  in  12. 

Third.  There  is  still  another  method  of  Solution. 
1st:  Since  12  apples  are  to  be 

T.TT  •i-.r»  ckJ  O*  THIRD  SOLUTION. 

divided,  we  write  12.    2d :   Since         -.  o  _^_  4.  _  q 
each  child  is  to  receive  4  apples,  we  —      _    . 

write  4  at  the  right  of  12.  3d: 
Since  4  apples  are  to  be  subtracted 
from  12  apples,  we  write  the  Sign 
Minus  between  12  and  4.  4th : 
Since  4  is  to  be  subtracted,  not  once 
only,  but  as  many  times  as  possible, 
we  write  a  dot  above  the  centre  of 
the  Sign  Minus,  and  another  below 
it,  to  show  this.  5th :  Since  we  wish  to  show  what 
number  of  times  this  result  equals,  we  write  the  Sign  of 
Equality  after  the  4.  6th :  Since  we  find  that  the  num- 
ber of  times  4  can  be  subtracted  from  12  equals  3  times, 
we  write  3  at  the  right  of  the  Sign  of  Equality.  7th : 
We  read  this  expression  thus :  12  divided  by  ^  equal  8. 

The  Sign  thus  made  by  changing  the  Sign  Minus  is 
named  the  Sign  of  Division. 


MENTAL  AND    WRITTEN  ARITHMETIC, 


99 


LESSON   LXVIII. 

By  Graded  Subtraction^  wg  find  how  many  times 
one  number  can  be  subtracted  from  another,  or  is  con- 
tained in  another. 

Division  is  a  short  method  of  performing  Graded 
Subtraction, 

The  Graded  Subtraction  Tables  on  pages  83  and  85 
can  be  thus  read  as  a 


DIVISION  TABLE, 


2  are  in 


3  are  in 


2  Once, 
4  2  Times, 
6  3  Times, 
8  4  Times, 
10  5  Times, 


12  6  Times, 
14  7  Times, 
16  8  Times, 
18  9  Times. 


3  Once, 

6  2  Times, 

9  3  Times, 

12  4  Times, 

15  5  Times, 


18  6  Times, 
21  7  Times, 
24  8  Times, 
27  9  Times. 


Mental  Exercises. 

14~2  =  ?       9-^3  =  ?       4-^2  =  ?  6-^3  =  ? 

18 -^3  =  ?     21 -^3  =  ?     12 -^2  =  ?  8-4-2  =  ? 

12-f-3  =  ?     10  -^2  =  ?     15-4-3:==?  16~2  =  ? 

6-^2  =  ?     18-4-2  =  ?     24~3  =  ?  27-4-3  =  ? 

1,  If  2  boys  can  sit  in  one  seat  at  school,  how  many 
seats  will  be  required  for  12  boys  ? 

Explanation.  Since  1  seat  is  required  for  2  boys, 
the  number  of  seats  required  for  12  boys  is  equal  to  the 
number  of  times  2  boys  are  contained  in  12  boys.  2  are 
in  12  6  times.  Therefore  6  seats  are  required  for  12 
boys. 

2,  18  apples  were  divided  among  a  company  of  boys. 
Each  boy  had  2  apples.    How  many  boys  were  there  ? 

3,  2  horses  make  1  span.  How  many  spans  will  16 
horses  make  ? 


ioo 


FIRST  LESSONS  IN 


LESSON   LXIX. 


THB  Tyro    GITJEJJSr  JSrUM'SB'RS    STAJVDIJVG 

It  often  happens  that  the  things  for  which  the  Divi- 
dend and  the  Divisor  stand  are  unlike  in  hind. 

Example.  Harry's  mother  gave  12  peaches  to  her  3 
children,  Albert,  Harry  and  Walter,  giving  each  the 
same  number.     How  many  peaches  did  each  receive  ? 

Explain" ATioi^.  The  Dividend 
is  12  peaches  ;  and  the  number 
S,  which  stands  for  hoys,  is  the 
Divisor.  1st:  We  take  3  peaches 
from  12  peaches,  and  place  them 
on  the  table,  in  a  row,  headed 
"  1st  3.''  Each  boy  has  one  of 
these  3  peaches.  2d:  Finding, 
by  Subtraction,  that  3  peaches 
are  contained  in  12  peaches  ^ 
times,  we  place  4  ^^ws  on  the 
table. 


SOLUTIOir. 

B7J  Subtraction. 

12  Peaches. 

3 

i6 

1st 

9 

cc 

3 

(C 

Sd 

6 

<6 

3 

iC 

Sd 

3 

i( 

_3 

<e 

J4h 

No  Peaches 

', 

MENTAL  AND    WRIT'L'PN  Au'lTBTlfMTXC,  :»01 

Each  boy  has  1  peach  in  each  row,  or  4  peaches 
in  all. 

But  we  can  consider  the  row5  running  crosswise  of 
these.  Then  each  boy's  peaches  will  stand  in  a  row. 
There  will  be  3  rows  ;  one  row  for  each  boy,  with  4 
peaches. 

In  this  Solution,  1st:  We  divide  the  Dividend,  12 
peaches,  into  parts,  each  having  3  peaches,  and  find  that 
there  are  4  parts,  or  rows.  2d :  We  then  consider  the 
rows  running  crosswise  of  the  first  set,  and  find  there 
are  3  parts,  or  rows,  with  4  peaches  in  each. 

We  divide  by  3  in  the  same  manner 
as  we  would  if  it  stood  iov  peaches.    In       solution. 
the  end,  however,  the  Divisor,  3,  is      ^y^^^^- 
made  to  stand  for  the  number  of  parts,        ^  Quottent. 
or  rows,  into  which  we  divide  the  Divi-      — 
dend;  and  the  Quotient,  4,  shows  the      ^^  Dividend, 
size  of  each  part  of  the  Dividend.    Hence, 

I^rinciple  1, 

When  the  Divisor  shows  the  kumber  of  parts  into 
which  the  Dividend  is  divided,  the  Quotient  shows  the 
SIZE  OF  EACH  PART  of  the  Dividend. 

Principle  2, 

When  the  Divisor  shozvs  the  size  of  each  of  the 
PARTS  into  which  the  Dividend  is  divided,  the  Quotient 

€h0WS  THE  NUMBER  OF  THE  PARTS. 
I^nHple  3, 

If  we  regard  both  the  Divisor  and  Quotient  in  any 
Example  in  Division,  one  of  them  tcill  show  the  num- 
ber of  parts  into  which  the  Dividend  is  divided,  and 
the  other  the  size  of  each  part. 


K)3  'WR^T  -LESSONS  IN 

LESSON   LXX, 

MUZTITI.ICATIOJ\r. 

Example  1.  Willie's  mother  gave  5  apples  to  each  of 
her  3  children.    How  many  apples  did  she  give  in  all  ? 

ExPLAiq-ATiON".  1st.  Our  Product, 
15,  shows  tlie  whole  number  of  ap-  solution. 

pies  given.     2d.  Our  Multiplier,  3,      5   Multiplicand. 
shows  the  number  of  parts,  or  groups,    _^  Multiplier, 
into  which  the  15   apples  were  di-    15   Product, 
vided.      3d.    Our    Multiplicand,   5, 
shows  the  size  of  each  of  the  parts,  or  groups,  into  which 
the  15  apples  were  divided,  in  giving  them  to  the  3 
children. 

0irisioj\r. 

Example  2.  Willie's  mother  divided  15  apples  among 
her  3  children,  giving  each  the  same  number.  How 
many  apples  did  she  give  each  child  ? 

T-.  -rrr    .  .  .  w  ^  ,1  80LUTI0K. 

Explanation^.  Writmg  15  as  the  k  n    t'    t 

Dividend,  and  3  as  Divisor,  and  pro-  3  Divisor 

ceeding  as  in  the  Solution  on  page  ^  Dividend. 
101,  the  Quotient  is  5  apples. 

Examining  these  two  Solutions,  we  observe : 

1st.  15 f  which  shoivs  the  i^umber  of  thiitgs  di- 
vided, is  the  Product  in  MultijMcation  and  the  Divi- 
dend in  Division. 

2d.  3f  tuhich  shows  the  kumber  of  parts  into 
which  15  things  are  divided,  is  the  Multiplier  in  Mul- 
tiplication and  the  Divisor  in  Division. 

3d.  89  which  shows  the  size  of  each  of  the  parts 
into  which  15  things  are  divided,  is  the  Multiplicand 
in  Multiplication  and  the  Quotient  in  Division. 

J4h.  The  SAME  THREE  NUMBERS  occur  in  both  Solutions, 


MENTAL  AND    WRITTEN  ARITHMETIC.  103 

INFERENCES, 

1st.  The  Pkoduct  in  Multiplication  may  he  taken  as 

DlVIDEJSTD. 

^d.  The  Multiplier  in  Multiplication  may  he  taken 
as  DiYisoR. 

Sd.  If  the  Product  he  taken  as  Dividend,  and  the  Mul- 
tijMer  as  Divisor,  the  Quotieot:  in  Division  will  he  the 
same  as  the  Multiplica^^d  in  Multiplication. 

Jfih.  If  the  Divisor  and  Quotient  be  multiplied 

TOGETHER,  THE  RESULT  WILL  EQUAL  THE  DIVIDEKD. 
Principle  4. 

Division  consists  in  finding  a  kumber  which,  wheit 

MULTIPLIED    BY    OKE     OF    TWO     GIVEN*    NUMBERS,    wHl 

give  a  Product  equal  to  the  other  given  numher. 

TESTING  TME  QUOTIENT. 

After  obtaining  our  Quotient,  we  can  always  test  it 
by  Diference  Jf.  Multiplying  it  by  the  Divisor,  if  the 
result  equals  the  Dividend  we  presume  that  our  Quo- 
tient is  the  true  one. 


LESSON   LXXI. 

Find  and  test  the  Quotients  in  the  following 
Exercises  for  the  Slate  and  Board. 
Quotients.     6       ????????       ? 
Divisors.    _5_3^^_2_3_2^^_3 

Dividends.  12     18     14     15     18     21     16    27     10     24 

Multiplication, 

Example  1.  Find  the  Product  solution. 

arising  from  multiplying  34  by  2.  34  Multiplicand. 

__                           TTT           1  i  •  1  ^  Jjluitvplier. 

Explanation.     We    multiply  —          ^ 

first  the  4  Ones,  and  then  the  3        ^^  Product. 

Tens,  by  2,  as  in  former  Examples, 

and,  writing  the  results  in  the  Product,  have  68. 


104  FIRST  LESSONS  IN 


Division, 


Example  2.  Find  the  Quotient  arising  from  dividing 
68  by  2. 

ExPLAKATiOK.  From  "Principle  solution. 

4/^  we  find  that  Division  consists,  in      34  Quotient. 
this  Example,  in  finding  a  number      _^  Divisor. 
which,  when  multiplied  by  2,  will       G8  Dividend. 
give  a  result  equal  to  68.  68  Test  Product. 

1st.  We  find  how  many  Tens  mul- 
tiplied by  2  will  give  the  6  Tens  of  the  Dividend.    Since 
3  Tens  multiplied  by  2  give  6  Tens,  we  write  3  Tens  in 
the  Quotient. 

2d.  Finding  that  4  Ones  multiplied  by  2  give  8  Ones, 
we  write  4  Ones  in  the  Quotient. 

3d.  Testing  our  Quotient^  we  multiply  34  by  2,  and 
write  the  result  68,  as  a  Test  Product^  below  the  Divi- 
dend. 

Find  and  Test  the  Quotients  in  the  following 

Exercises  eor  the  Slate  an^d  Board. 


Quotients. 
Divisors. 

? 
2 

? 
2 

X. 

? 

2 

? 

2 

? 

2 

? 

2 

? 
2 

Dividends. 

42 

64 

46 

68 

48 

88 

68 

Test  Producti 

!.  42 

IL 

.... 

.... 

Divisors. 

? 
3 

V 

3 

? 

3 

? 
3 

? 
3 

? 
3 

? 
•3 

Dividends. 

63 

66 

39 
III. 

69 

93 

99 

369 

468  --  2 
639  --3 

684  -f 
306^ 

-2 
-3 

2,468 
9,630 

-f-  2 
-v-3 

68,468 
60,936 

-^  2 
-i-3 

MENTAL  AND    WRITTEN  ARITHMETIC, 


105 


LESSON   LXXII. 

M772.  TI'PI.ICA^TIOJV. 


Example  A.  Multiply  376  by 


376 

__2 

12 
14 
6 

752' 


SOLUTIOiy. 

Multiplicand. 
Multiplier. 

Partial 
Products. 

Product. 


2. 

ExPLAN^ATiOK.  Multiplying  6 
Ones,  then  7  Tens,  and  lastly  3 
Hundreds,  separately  by  2,  and 
writing  the  Partial  Products  and 
adding  them,  we  have  752  for  our 
final  Product. 

^irisiojv-. 

Example  B.  Divide  752  by  2. 

ExPLAKATiois".  1st.  Our  Hundreds 
in  the  Quotient  cannot  be  more  than 
3,  since  4  Hundreds  multiplied  by  2 
would  give  8  Hundreds.  Hence  we 
write  3  Hundreds  in  the  Quotient. 
Multiplying  3  Hundreds  by  2,  we 
write  the  Product,  6  Hundreds,  under 
the  7  Hundreds.  Drawing  a  line  below 
the  6,  we  subtract  6  from  7,  and  have 
1  Hundred  remaining. 

2d.  We  now  bring  down  the  5  Tens  of  the  Dividend ; 
and,  writing  5  at  the  right  of  the  1  Hundred,  have  1 
Hundred  and  5  Tens,  or  15  Tens,  for  a  new  Partial 
Dividend.  Proceeding  to  divide  this  by  2,  we  see  that 
we  cannot  have  more  than  7  Tens  in  the  Quotient. 
Writing  7  Tens  in  the  Quotient,  and  multiplying  them 
by  2,  we  write  the  Product,  14  Tens,  under  the  15  Tens. 
Subtracting,  we  have  1  Ten  remaining. 

3d.  Lastly,  we  bring  down  the  2  Ones  of  the  Divi- 
dend ;  and,  writing  2  at  the  right  of  the  1  Ten,  have  1 
Ten  and  2  Ones,  or  12  Ones.     Dividing  12  Ones  by  2, 


376  Quotient. 
2  Divisor, 

752  Dividend. 
6_ 

15 
14 

12 
12 


106  FIRST  LESSONS  IN    . 

we  write  6  Ones  in  the  Quotient.  Multiplying  6  Ones 
by  2,  and  subtracting,  there  is  no  Remainder,  Hence 
our  entire  Quotient  is  376. 

TIi:STING  THE  QXTOTIENT. 

To  I'est  our  Quotient  we  would  multiply  it  by  the 
Divisor.  The  work  would  be  the  same  as  in  Example 
Ar 

From  our  work  in  both  Examples,  we  obserye,  1st : 
The  Partial  Products  in  both  are  6  Hundreds,  14  Tens, 
and  12  Ones;  but  they  stand  in  contrary  order.  In 
Example  A,  we  added  these,  and  obtained  752.  In 
Example  B,  we  subtracted  them  from  752,  in  a  reverse 
order,  and  had  no  Remainder,  2d :  In  Example  A,  we 
multiplied  together  two  numbers,  376  and  2,  and  obtained 
752  for  a  Product ;  and  in  Example  B  we  divided 
this  Product  by  one  of  the  numbers,  2,  and  obtained,  for 
a  Quotient,  the  other  number,  376.  Hence,  we  draw  the 
following 

INFERENCE, 

DiYisiOK  is  precisely  the  reyekse  of  Multiplica- 

TIOIS". 

Principle  5. 

If  the  Product  of  two  numbers  be  diyided  by  oke  of 
them,  the  Quotient  will  be  the  other. 

Find  and  Test  the  Quotients  in  the  following 

Exercises  eor  the  Slate  ajstd  Board. 

I. 

Quotients,        ?????? 

Divisors,      _2  _3        _2        __3        _2  3 

Divide7ids,   312        435        538        768        756        867. 

II. 
5,871  -^  3         9,786  -:-  2         7,467  ~  3         9,536  -r-  2 
3,976  -^  2         8,568  -^  3         5,796  -^  2         4,125  -r-  3 
7,641  -T-  3         5,876  -^  2         8,085  ~  3         1,974  -^•  2 


MENTAL  AND    WRITTEN  ARITHMETIC. 


107 


LESSON   LXXIII. 

The  three  numbers  will  be  precisely  alike,  when  ob- 
tained, in  both  Examples  of  each  Pair  of  the  following 


EXEKCISES   FOR  THE   SlATE   AND   BOARD. 


Multiplication 

1.  \ 

Division 


59,758 

2 


l  119,516 


789,675 
3 


2,369,025 


S.\ 


376,586 
2 


l  753,172 


By  Principle  5  we  change  our  Multiplication  Tables 
into  two  sets  of  Division  Tables ;  thus 


2    lare    2, 

3     2 

1  Time, 

It   2 

9  Times, 

2     2  are    4, 

2      4 

2  Times, 

2   4 

2  Times, 

2     3  are    6, 

2      6 

3  Times, 

3      6 

2  Times, 

2^4  are    8, 

2g    8 

4  Times, 

4      8 

2  Times, 

2  1  5  are  10, 

2X10 

5  Times, 

5I1O 

2  Times, 

2  "^  6  are  12, 

2«12 

6  Times, 

6|12 

2  Times, 

2     7  are  14, 

2    14 

7  Times, 

7    14 

2  Times, 

2     8  are  16, 

2    16 

8  Times, 

8    16 

2  Times, 

2     9  are  18, 

2    18 

9  Times, 

9    18 

2  Times. 

3    1  are    3, 

3      3 

1  Time, 

1|3 

3  Times, 

3     2  are    6, 

3      6 

2  Times, 

2      6 

3  Times, 

3     3  are    9, 

3      9 

3  Times, 

3      9 

3  Times, 

3  ^  4  are  12, 

3gl2 

4  Times, 

4    12 

3  Times, 

3  1  5  are  15, 

3^15 

5  Times, 

5ll5 

3  Times, 

3  "^  6  are  18, 

3«18 

6  Times, 

6|18 

3  Times, 

3     7  are  21, 

3    21 

7  Times, 

7    21 

3  Times, 

3     8  are  24, 

3    24 

8  Times, 

8    24 

3  Times, 

3     9  are  27, 
8 

3    27 

9  Times, 

9    27 

3  Times. 

108  FIRST  LESSONS  IN 

LESSON   LXXIV. 

The  equation  5  x  3  =  15  formed  by  Multiplication^ 
we  name  an  JEquation  hy  Multiplication. 

Since  the  equation  15  -r-  3  =  5  is  formed  by  Division^ 
we  wiU  name  it  an  JEquation  by  Dlmsion. 

From  three  numbers  such  that  no  two  of  them  are 
equal,  and  the  greatest  equals  the  Product  of  the  two 
others,  we  can  form  two  Equations  hy  Multiplication^ 
by  the  Principle  given  on  page  93,  and  also  two  Equor 
tions  by  Division^  by  Principle  5  in  Division. 

To  Form  two  Equations  by  Multiplication: 

KULE. 

I.  Write  the  two  smaller  numbers^  with  the  Sign  x 
between  them,  for  the  First  Member  of  an  Equation. 

II.  Write  the  greatest  number  for  the  Second  Member, 
placing  the  Sign  =  between  the  Members. 

III.  Form  the  second  Equation  from  the  first,  by 
changing  the  places  of  the  two  smaller  numbers. 

To  Form  two  Equations  by  Division: 

EULE. 

I.  Write  the  greatest  number,  and  after  it  one  of  the 
other  numbers,  placing  the  Sign  -r-  between  them,  for  the 
First  Member  of  an  Equation. 

n.  Write  the  remaining  number  for  the  Second  Mem- 
ber, placing  the  Sign  =  between  the  Members. 

in.  F(yrm  the  second  Equation  by  Division  from  the 
first,  by  changing  the  places  of  the  two  smaller  numbers. 

Form  4  Equations  from  each  of  these  sets  of  numbers : 
2,  7  and  14;  6,  3  and  18  ;  2,  8  and  16;  3,  7  and  21 ; 
9,  2  and  18;  3,  8  and  24 ;  6,  2  and  12 ;  9,  3  and  27 ; 
5,  3  and  15 ;    4,  3  and  12 ;    2,  5  and  10 ;    4,  2  and  8. 


MENTAL  AND    WRITTEN  ARITHMETIC.  109 

LESSON   LXXV. 

In  the  Equation  2  x  3  =  6,  if  6  were  erased,  it  could 
be  found  by  7nulti2Jlying  together  2  and  3 ;  if  2  were 
erased,  we  could  find  it  by  dividing  6  by  3;  and  if  3 
were  erased,  it  could  be  found  by  dividing  6  by  2. 
Hence, 

To  Replace  a  Number  in  an  Equation  by 
Multiplication: 

KULE. 

L  If  the  Second  Member  he  missing,  find  it  ly  Multi- 
plying together  the  tioo  numbers  in  the  First  Member. 

II.  If  either  number  in  the  First  Member  be  missing, 
find  it  by  Dividin^g  the  secokd  member  by  the  other 
NUMBER  in  the  First  Member. 

Exercises  for  the  Slate  and  Board. 

3x?  =  18?x5  =  15     8x2  =  ?       6x3  =  ? 
?x3  =  12    7x?==14     7x3  =  ?       7x?=21 
3  X  8  =  ?      3  X  ?  =  27      ?  X  6  =  18    2  X  ?  =  16 

As  we  see  at  the  right, 
i\iQ  three  numbers  mi]iQ^        Quotient.      7  Multiplicand. 
Solution  of  an  Example        Divisor.     _3  Multiplier. 
in   Multiplication    and        Dividend.  21  Product. 
one  in  Division  are  the 
same,  and  stand  in  the  same  order.    Hence, 

To  FIND  ANY'  NUMBER  IN  AN  EXAMPLE  IN  MULTI- 
PLICATION OR  Division,  when  missing: 

EULE. 

I.  If  EITHER  OF  THE  FIRST  TWO,  Or  Smaller  numbers, 
be  missing,  find  it  by  dividing  the  third,  or  greatest, 
by  the  smaller  number  tohich  is  given.  - 

II.  If  the  THIRD,  or  greatest  number,  be  missing,  find  it 
by  multiplying  together  the  two  others. 


110 


FIRST  LESSONS  IN 


Exercises  por  the  Slate  ai^d  Board. 
5        ?         8        ?         ?         3        3        ?  -     ?       9 

10       14       16       18       12       15       18       21       24     27 


LESSON   LXXVI. 

We  now  place 
the  3  liorizon- 
tal  rows  of 
cubes,  on  page 
85,  over  an- 
other row. 
The  portion 
now  added  to 
the  Table  is 
formed  in  the  same  manner  as  the  portion  then  made. 

From  the  Multiplication  Table  given  below,  and  also 
from  each  one  hereafter  given,  make  a  second  Table  by 
changing  the  places  of  the  figures  in  the  first  and  second 
columns;  thus:  1  time  4  is  4;  2  times  4  are  8;  3  times 
4  are  12 ;  &c.,  &c. 


/    /    r 

W~ 

L. 

m^ 

-   \ 

r  f"  ^ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2 

4 

6 

8 

10 

12 

14 

16 

18 

3 

6 

9 

12 

15 

18 

21 

24 

27 

J4 

8 

12 

_I6_ 

20 

24 

28 

32 

^ 

TABLE, 

DIVISION 

TABLES, 

4t    1  are    4, 

4      4 

1  Time, 

1 1   4  4  Times, 

4    2  are    8, 

4      8 

2  Times, 

2'      8    4  Times, 

4    3  are  12, 

4     12 

3  Times, 

3     12    4  Times, 

4  «  4  are  16, 

4|16 

4  Times, 

4    16    4  Times, 

4|  5  are  20, 

4  120 

5  Times, 

5I2O    4  Times, 

4^  6  are  24, 

4''24 

6  Times, 

6|24    4  Times,  I 

4     7  are  28, 

,  4    28 

7  Times, 

7    28    4  Times, 

4    8  are  32, 

4    32 

8  Times, 

8    32    4  Times, 

4    9  are  36. 

4    36 

9  Times, 

9    36    4  Times. 

MENTAL  AND    WRITTEN  ARITHMETIC,  111 

The  same  3  columns  of  figures  occur  in  all  the  3  Ta- 
bles on  the  preceding  page. 

Mental  Exercises. 

I. 

9x4  =  ?       5x?=2Q    7x?==28  8x?=32 

6x?  =  18     ?x4  =  24    ?x4r=:28  9x?  =  36 

?x8  =  32     4x?=28     6x?=24  ?x4  =  36 

II. 

16-T-4  =  ?     21-4-7  =  ?     28 -f-?  =7  ?  -r  6  =  4 

21  ^9  =  r^    20-f-4  =  ?     16-T-4  =  ?  ?-4-4  =  4 

24-v-4  =  ?       ?-T-4  =  9       ?-f-8  =  4  36-r-?=4 


LESSON   LXXVII. 

MUL  TI^ZICA  TIOJV. 

587,936 
4 

I. 
759,864               434,234 
4                          9 

II. 

342,344 
8 

241,243 

7 

423,132               214,324 
6                          5 

879,567 
4 

III. 
Multiply  414,342  by  4;     by  5;     by  6;     by  7;     by  8; 
Multiply  340,424  by  4 ;     by  5 ;     by  6 ;     by  7 ;     by  8. 

^irisiojsr. 
I. 

4  3  4  8 


7,849,896  8,626,587  4,787,104  2,738,752 

II. 
7  6  5  4 


1,688,701  2,538,726  7,171,060  9,276,872 


112 


FIB8T  LESSONS  IN 


LXXVIII. 


LESSON 

It  should  be 
noticed  that 
the  first  part 
of  the  Tables 
given  below, 
extending  t  o 
the  line  com- 
mencing "  5 
times  4  are 
20/'  has  been 
given  in  the  preceding  Tables.  The  new  part  is  printed 
in  the  Italic  type. 

MTTLTIPJLICATION 
TABLE, 


f 

r 

L-^K"  fc^ 

fc 

c— 1 

^  ^  ^ 

1 

2 

3 

4. 

5 

6 

7 

8 

9 

2 

4 

6 

8 

10 

12 

14 

16 

18 

3 

6 

9 

12 

15 

18 

2i 

24 

27 

4 

8 

12 

16 

20 

24 

28 

32 

36 

js 

10 

15 

20 

25 

30 

35 

40 

45  [ 

5 

1 

are 

5, 

5 

2 

are 

10, 

5 

3 

are 

15, 

^t 

4 

are 

20, 

^1 

5 

are 

25, 

6^ 

(5 

are 

SO, 

5 

7 

are 

85, 

5 

8 

are 

JfO, 

6 

9 

are 

J^. 

5 

5 
5 
5j 


5 
10 
15 
20 


DIVISION  TABZBS, 

1  Time,  l| 

2"* 
3 
4 


5 
10 
15 
20 


5  I  25 
5""  SO 


S5 

JfO 


5  $25 
6tS0 

7  S5 

8  Jfi 

9  JfB 


5  Times, 
5  Times, 
5  Times, 
5  Times, 
5  Times, 
5  Times, 
5  Times, 
5  Times, 
5  Times. 


2  Times, 

3  Times, 

4  Times, 

5  Times, 

6  Times, 

7  Times, 

8  Times, 

9  Times, 

"Write  one  more  Equation  by  Multiplication  and  two 
by  Division  from  each  of  the  following 

Equations. 

x7  =  35  6x5  =  30 
x8  =  32     9x4  =  36 

MEi^TAL  EXEKCISES. 
I. 

X  ?  =  36  5  X  ?  =  30 
X  ?  :=  40  9  X  ?  =  27 
X  5  ==  25     ?  X  9  =3  45 


24 
40 


=  ? 
=  36 
=  40 


4 

8 

? 


4 

X 

7 

r:: 

28 

9 

X 

5 

= 

45 

5 

X 

9 

^ 

? 

5 

X 

? 

= 

45 

7 

X 

p 

= 

35 

MENTAL  AND    WRITTEN  ARITHMETIC, 


113 


II. 


35  ^5  =  ?     ?^7 
30  ^  5  =  ?    40  ^  8 

=  3 

=  ? 

36  ^ 

28  i- 

4  = 

•  7  = 

?     36  -f-  9  =  ? 
?       ?  -^  9  =  5 

5         ?         ?         9 

7         7        4         5 

? 
9 

III. 

7 
? 

5 
8 

5         ?         9 

?        5          ? 

?       35      28         ? 

45 

35 

? 

30      40       45 

LESSON  LXXIX. 

Exercises  for  the  Slate  akd  Board. 


MuUipUcation. 

1.  Multiply  874,596  by  2; 

by  3; 

by  4; 

by  5. 

2.  Multiply  746,789  by  2 ; 

by  3; 

by  4; 

by  5. 

3.  Multiply  876,598  by  2; 

by  3; 

by  4; 

by  5. 

A.  Multiply  715,829  by  2; 

by  3; 

by  4; 

by  5. 

5.  Multiply  769,854  by  2; 

by  3; 

by  4; 

by  5. 

I>ivUU>n, 

L  Divide  114,840  by  2; 

by  3; 

by  4; 

by  5. 

2.  Divide  689,040  by  2 ; 

by  3; 

by  4; 

by  5. 

3.  Divide  803,880  by  2 ; 

by  3; 

by  4; 

by  5. 

Jf.  Divide  344,520  by  2; 

by  3; 

by  4; 

by  5. 

5.  Divide  107,640  by  2; 

by  3; 

by  4; 

by  5. 

Multiplication  at  Sight 

, 

4x4       4x5       6x5 

4x7 

6x4 

7x4 

5x5       4x6       5x4 

7x5 

5x6 

3x9 

3x8       5x8       8x4 

9x3 

8x3 

4x9 

Division  at  Sight. 

6^2         5-4-5         8-f-4 

10-4-2 

10-4-5 

14 -=-3 

8-=-2    20-f-5      9-f-3 

25  -^  5 

12-4-2 

1^-4-6 

15-4-5    16  -T-  4    16-1-8 

12  H-  3 

12-4-4 

18^2 

20-4-4    15  -^  3    14  -^  7 

16-4-2 

18-4-3 

21-4-3 

114 


FIRST  LESSONS  IN 


LESSON   LXXX. 


The  four 
Tables  given 
below  should 
be  thoroughly 
learned.  They 
can  all  be  read 
from  the  Table 
of  cubes  shown 
at  the  right. 
The  new  part 
now  added  is 
printed  in  Ita- 
lic type. 
MULTIBTj  tcation 

TABL£J. 


f 

f-    ^  ^ 

L 

1= 

^-      \ 

^     ^     \ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2 

4 

6 

8 

10 

12 

14 

16 

18 

3 

6 

9 

12 

15 

18 

21 

24 

27 

4 

8 

12 

i6 

20 

24 

28 

32 

36 

5 

10 

15 

20 

25 

30 

35 

40 

45 

6 

12 

ii 

24 

30 

3£ 

42 

48 

54^ 

DIVISION  TABIBS, 


6,1  5  are  30, 
^^^are.^^, 
6  1l^x^Ji2, 
6  8  are  4^^, 
6    Pare  54. 


6    1  are    6,       6      6  1  Time,  1  "J 

6    2  are  12,        6    12  2  Times,  2*" 

6    3  are  18,        6    18  3  Times,  3 

6  *  4  are  24,        6 1  24  4  Times,  4 

6  1^  30  5  Times,  5 1 

e'^Se  ^  Times, 

6    42  7  Times, 

6    JfB  ^  Times, 

6  ^5Jf,  9  Times, 
Write  three  other  Equations  from  each  of  the  follow- 
ing 


8 
9 


6  ©Times, 

12    6  Times, 

18    6  Times, 

24    6  Times, 

30    6  Times, 

36    6  Times, 

6  Times, 

6  Times, 

6  Times. 


JfS 
54 


X  7  =  42 
X  8  =40 


Equations. 

6x8  =  48     6x9  =  54    5x6  =  30 
7x5  =  35     9x4  =  36     9x5  =  45 


?  X  7  =  42 
9  X  ?  =  54 


Mei^tal  Exekcises. 
I. 

6x8  =  ?       7  X  ?  =  42 
?  X  8  =  48      ?  X  6  =  48 


6  X  ?  =  36 
?  X  6  =  54 


MENTAL  AND    WRITTEN  ARITHMETIC. 


115 


II. 

36-T-6  =  ?     54-T-?  =  6     25 -^5  =  ?     30-^?  =  5 


III. 

? 

? 

7 

8 

e 

? 

8 

? 

5 

9 

6 

e 

? 

? 

9 

6 

6 

9 

? 

6 

36 

42 

42 

48 

48 

54 

? 

54 

35 

? 

LESSON   LXXXI. 

MirZ  TJTZICil  TIOJV. 


1,  Multiply  875,964  by  2 

;    by  3^ 

by  4; 

by  5 

,   bye. 

2,  Multiply  587,958  by  2 

;    by  3 

.    by  4; 

by  5 

bye. 

S.  Multiply  978,547  by  2 

;     by  3 

;    by  4; 

by  5 

;    bye. 

^.  Multiply  849,756  by  2 

;    by  3 

;    by  4; 

by  5 

;   bye. 

5.  Multiply  354,623  by  5 

;    by  6 

;    by  7; 

by  8 

,    by  9. 

6,  Multiply  560,365  by  5 

;     by  6 

;    by  7; 

by  8 

;    by  9. 

7.  Multiply  465,324  by  5 

;    by  6 

dsion. 

;    by  7; 

by  8 

;    by  9. 

1,  Divide  3,232,740  by  2  ; 

by  3; 

by  4; 

by  5 

;    bye. 

2.  Divide  4,765,140  by  2  ; 

by  3; 

by  4; 

by  5 

;  bye. 

S,  Divide  1,854,240  by  2  ; 

by  3; 

by  4; 

by  5 

;  bye. 

Jf.  Divide  1,683,840  by  2  ; 

by  3; 

by  4; 

by  5 

;  bye. 

J.  Divide  2,918,520  by  2  ; 

by  3. 

by  4; 

by  5 

;  bye. 

MultiplU 

nation  at 

Sight. 

4x5       6x4       8xi 

5       3  > 

c  9       5 

X  8 

7x5 

6x6       7x6       6x 

9       5  > 

<  6       9 

X  5 

6x8 

6x7       8x6       4x 

5       5  > 

<  9       9 

X  6 

6x5 

Division  at  Sight, 

42  -^  6  36  -f-  6  45  -f-  9  54  -^  6  48  -^  8  40  -=-  8 

35  -T-  7  54  -^  9  48  -^  6  42-7  30-4-6  45-^-5 

36  -J-  9  35  -^  5  30  -^  5  32  ~  4  40  -^  5  25-5 


116 


FIRST  LESSONS  IN 


LESSON   LXXXII. 


>    >-    r   *" 

t. 

fc 

^_     ' 

=      ^     \ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2 

4 

6 

8 

ID 

12 

14 

16 

18 

3 

6 

9 

12 

15 

18 

21 

24 

27 

4 

8 

12 

16 

20 

24 

28 

32 

36 

5 

10 

15 

20 

25 

30 

35 

40 

45 

6 

12 

18 

24 

30 

36 

42 

48 

54 

7 

14 

21 

28 

35 

42 

49 

56 

63 

MUZTIPZICA  TION 
TABLE. 


7    1  are    7, 

7      7 

7    2  are  14, 

7    14 

7    3  are  21, 

7    21 

7  *  4  are  28, 

7|28 

7 1  5  are  35, 

7^35 

7^  6  are  42, 

7''42 

7         l^XQJfiy 

7    49 

7    S^reSe, 

7    56 

7    9  are  63, 

7    68 

DIVISION 

1  Time, 

2  Times, 

3  Times, 

4  Times, 

5  Times, 

6  Times, 

7  Times, 

8  Times, 
P  Times, 


TABIES, 

2"  14 

3  21 

4  28 
5 1  35 

6  I  42 

7  4^ 
<9  ^^ 
9    68 


7  Times, 
7  Times, 
7  Times, 
7  Times, 
7  Times, 
7  Times, 
7  Times, 
7  Times, 
7  Times. 


Write  three  others  from  each  of  the  following 

Equations. 

7x8  =  56      8x6  =  48      6x9=54 
35^7  =  5      42-^7  =  6      30-f-6=5 


7  X  9  =  63 
36  ~  9  =  4 


7  X  ?  =  42 
?  X  7  =  63 


Mektal  Exercises. 
I. 
7x?  =  56     7x?  =  63     7x?  =  49 
?x7  =  56     ?x9=63     9x7  =  ? 


MENTAL  AND  WRITTEN  ARITHMETIC.  117 

11. 

36  ^  6  =  ?     56  -^  ?  =  8     42  -f-  ?  =  6    63  -^  ?  =  9 

49  ^7  =  ?     63_^7^p        ?,^7=7       p.^^.^^ 
49_^?=:7       p_^g^7     5g^P^7      p^9^7 

HI. 

?  9        7         9  ?  ?  7  ?  6 

_8_?9_?_7__7_?J_? 
56        54        ?        63        56        49        63        48        42 


LESSON   LXXXIII, 
Exercises  for  the  Slate  and  Board. 


Multiplication, 

795,869  X  7 

978,549  X  7 

895,987  X  7 

536,754  X  8 

765,467  X  8 

657,327  X  8 

534,276  X  9 

374,657  X  9 

534,756  X  9 

978,679  X  7 

576,754  X  8 

Division, 

321,567  X  9 

5,984,678  -^  7 

9,854,789  -^  7 

6,897,583  --  7 

5,416,040  H-  8   . 

5,174,016  -4-  8 

4,996,504  -V-  8 

6,395,148  -T-  9 

3,784,878  h-  9 

1,918,575  -f-  9 

9,546,327  -f-  7 

3,715,616  -4-  8 

5,982,039  -V-  9 

Divide  544,320  by 

4;           by  5; 

by  6;           by  7. 

Divide.5,987,520by  4;        by  5; 

by  6;           by  7. 

Multiplication,  at  Sight, 

5x6       6x6 

5x8       8x4 

7x4       4x7 

7x7       7x8 

7x9       8x7 

9x7       6x7 

5x7       7x6 

6x5       5x9 

Division  at  Sight. 

7x5       5x5 

35  -T-  7     42  -^  7 

63  -f-  9     56  -^  7 

42-6    49  -f-  7 

54  ^  6     30  -f-  5 

48  H-  6     36  ^  6 

30  -^  6     48  -^  8 

56  -^  8     63  ^  7 

25  -^  5    54  -r^  9 

35  -^  5     28  -f-  7 

118 


FinsT  lbsso:ns  m 


LESSON  LXXXIV. 

f    r    r    fc.  ■ 

te 

^-      \ 

^   P  ^ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2 

4 

6 

8 

10 

12 

14 

16 

18 

3 

6 

9 

12 

15 

18 

21 

24 

27 

4 

8 

12 

16 

20 

24 

28 

32 

36 

5 

iO 

15 

20 

25 

30 

35 

40 

45 

6 

12 

18 

24 

30 

36 

42 

48 

54 

7 

14 

21 

28 

35 

42 

49 

56 

63 

k 

16 

24 

32 

40 

48 

56 

64 

7Z^ 

MUZTIPLICA  TION 
TABLE, 


DIVISION  TABLES, 


8     1  are    8,  8      8  1  Time, 

8    2  are  16,  8    16  2  Times, 

8    3  are  24,  8    24  3  Times, 

8^4  are  32,  8|32  4  Times, 

8|  5  are  40,  8*^40  5  Times, 

8^  6  are  48,  8*48  6  Times, 

8    7  are  56,  8    56  7  Times, 

8    8  are  6 J/,,  8    GJf,  8  Times, 

8    9  are  72,  8    72  9  Times, 

Write  three  others  from  each  of  the  following 

Equations, 

6  X  8  =  48    7  X  9  =  63    9  x  8  =  72    7  x  8  =  56 

Mektal  Exeecises. 
I. 

7x8  =  ?      8x?=:56      ?x6=:48     8x?  =  64 
9x8=r?      9x?=:63      8x?  =  72     ?x9  =  7'2 


1-^8  8  Times, 

2"'l6  8  Times, 

3  24  8  Times, 

4  32  8  Times, 
5 1  40  8  Times, 

6  I  48  8  Times, 

7  56  8  Times, 

8  64.  ^  Times, 

9  72  8  Times. 


MENTAL  AND    WRITTEN  ARITHMETIC. 


119 


64  4-  8  =  ?  56  ^  r 

?  -^  7  =  8  73  H-  8 

II. 

=  7  73  -4- 

=  ?    ?-r 

9  = 
9  = 

? 
8 

? 
56 

-T-8  =  8 
-=-?  =  8 

7    ?    8    9 
7    8?? 
?   56   64   73 

III. 

?   8 
9   9 

63   ? 

7 
? 

56 

? 
8 

73 

?   8 
8   8 

64   ? 

LESSON   LXXXV. 
Exercises  for  the  Slate  and  Board. 


JHultiplication, 

785,968  X  8  978,786  x  8 

874,569  X  8  678,587  x  9 

854,867  X  9  758,675  x  9 

1.  Multiply  859,756  by  5 ;  by  6 ; 

2.  Multiply  596,873  by  5 ;  by  6 ; 
S.  Multiply  584,762  by  4 ;  by  7 ; 
Jf,.  Multiply  378,578  by  4 ;  by  7 ; 


687,952 
789,872 
245,439 


jyivision. 

598,648  -^  8 
759,132  -r-  9 
753,588  -T-  9 


1.  Divide  164,304  by  4; 

2.  Divide  332,472  by  4; 
S.  Divide  349,440  by  4 ; 
^.  Divide  903,504  by  2 ; 
5.  Divide  910,680  by  3 ; 


by  6; 
by  6; 
by  6; 
by  3; 
by  4; 


8x9       7x8 


Multiplieation  at  Sight. 

8x8       7x9 


879,587  X  8 
768,576  X  9 
795,879  X  8 

by  7;        by  8. 

by  7;        by  8. 

by  8 ;        by  9. 

by  8;        by  9. 

987,688  -T-  8 
653,445  -^  9 
579,568  -^  8 


by  7 
by  7 
by  7 
by  4 
by  5 


by  8. 
by  8. 
by  8. 
by  8. 
by  8. 


9x8       6x8 


Division  at  Sight. 

63  -^  9     64-4-8     72  4-  8     56  -^  8     72  4-  9    48-4-8 
1.9-7     56  4-  7     48  4-  6    63  4-  7    40  4-  5    45  -^  9 


120 


FIRST  LESSONS  IN 


LESSON   LXXXVI. 

f   r    ^ 

^-  t_  1 

fe   fct-   %,.  ^ 

fe^    \ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2 

4 

6 

8 

10 

12 

14 

16 

18 

3 

6 

9 

12 

15 

18 

21 

24 

27 

4 

8 

12 

16 

20 

24 

28 

32 

36 

5 

10 

15 

20 

25 

30 

35 

40 

45 

6 

12 

18 

24 

30 

36 

42 

48 

54 

7 

14 

21 

28 

35 

42 

49 

56 

63 

8 

16 

24 

32 

40 

48 

56 

64 

7Z 

m 

18 

27 

36 

45 

54 

63 

72 

M 

TjLBLE. 

DIVISION  TABLES. 

9    1  are    9, 

9      9 

1  Time, 

l|    9  9  Times, 

9    2  are  18, 

9    18 

2  Times, 

2"*  18    9  Times, 

9    3  are  27, 

9    27 

3  Times, 

3    27    9  Times, 

9  «>  4  are  36, 

9-36 

4  Times, 

4-S  36    9  Times, 

9|  5  are  45, 

9  145 

5  Times, 

5  1  45    9  Times, 

9^  6  are  54, 

9    54 

6  Times, 

6    54    9  Times, 

9    7  are  63, 

9    63 

7  Times, 

7    63    9  Times, 

9    8  are  72, 

9    72 

8  Times, 

8    72    9  Times, 

9    QaieSL 

9  are  in 

81 

9  Times. 

Mektal  Exercises 

8  X  ?  =  56 

?  X  9  = 

:    63       8    X 

?  = 

72     9x9  =  ? 

?  X  9  =  72 

7  X  ?  = 

63     9  X 

7  = 

?       9x8=? 

8x8  =  ? 

9  X  ?  = 

81     ?  X 

8  = 

56     ?  X  9  =  81 

MENTAL  AND    WRITTEN  ARITHMETIC, 


121 


II. 

72  4-8  =  ?      81-^-9  =  ?      ?-^8  =  8        ?h-9  =  9 


? 
8 


III. 

9         ? 
9       _9 

? 


8 

? 


? 

8 


63     72       63       72 


81 


72       81      64 


ft    ^'  ^  ■ 

k-. 

fc 

^  ■  t 

t^  ) 

^  \ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2 

4 

6 

8 

10 

12 

14 

16 

18 

3 

9 

12 

15 

18 

21 

24 

27 

4 

16 

20 

24 

28 

32 

36 

5 

25 

30 

35 

40 

45 

6 

36 

42 

48 

54 

7 

49 

56 

63 

8 

64 

72 

J  9 

_ 

8l[ 

By  comparing  the  above  Table  with  that  given  on  the 
preceding  page,  it  will  be  seen  that  the  numbers  omitted 
in  forming  this  Table  from  that  are  the  same  as  those 
retained.    Hence  they  were  unnecessary. 

In  reading  this  Table  we  find  our  Multiplier,  or  Di- 
visor, in  the  upper  horizontal  row  whenever  it  is  greater 
than  the  Multiplicand,  or  Quotient. 
6 


122  FIRST  LESSONS  IN 


LESSON    LXXXVII. 

Example  A.  Multiply  964  by  3. 

ExpLA:N^ATio:Nr.  1st.  The  first  Solution  is  iu  the  form 
heretofore  used,  the  Partial  Products  being  written. 

2d.  In  the  second  Solution  the  Par- 
tial Products  are  not  written  out  in         first  solution. 
full.     We  first  multiply  4  Ones  by  3,  964 

and,  obtaining  12  Ones,  or  1  Ten  and  3 

2  Ones,  for  a  Partial  Product,  write 

the  2  Ones,  and  retain  the  i  Ten  in 

the  mind.     N^ext,  we  multiply  the  6 

Tens  by  3,  and,  obtaining  18  Tens  for  2,892 

a  Partial  Product,  we  add  to  these  the 

1  Ten  retained  in  the  mind,  and  have        second  solution. 

19  Tens,  or  1  Hundred  and  9  Tens.  964 

We  write  the  9  Tens  in  the  Product,  r 

and  retain  the  1  Hundred  in  the  mind.  2,892 

Finally,  multiplying  9  Hundreds  by  3, 
we  have  27  Hundreds  for  a  Partial  Product,  to  which 
we  add  the  1  Hundred  retained,  and  write  the  Sum,  28 
Hundreds,  in  the  Product.     Our  final  Product  is  2,892 ; 
the  same  as  in  the  first  Solution. 

These  2  methods  of  Multiplication  differ  in  this  only ; 
in  the  first  the  Partial  Products  are  written,  while  in 
the  second  only  the  final  Product  is  written.  The  writ- 
ten work  in  the  second  is  much  shorter  than  in  the  first. 

DEFINITIONS, 

1,  JLong  Multiplication  is  the  method  of  Mul- 
tiplication used  where  we  obtain  the  final  Product  by 
writing  out  the  Partial  Products  and  finding  their 
Sum. 


MENTAL  AND    WRITTEN  ARITHMETIC. 


123 


2,  Short  Multiplication  is  the  method  of  MuU 
tijjUcation  used  where,  after  writing  the  Multiplicand 
and  Multiplier,  we  write  the  final  Product  only,  perform- 
ing the  rest  of  the  work  mejitally. 

Find  the  Products  by  Short  Multiplication  in  these 

Exercises  for  the  Slate  ai^^d  Board. 

879,567  X  2  543,763  x  3                825,046  x  4 

342,354  X  5  243,652  x  6                105,824  x  9 

243,054  X  6  540,324  x  8                134,564  x  9 

536,726  X  6  947,540  x  8          '      302,132  x  9 


LESSON   LXXXVIII. 

Example  B.  Divide  2,892  by  3. 

ExPLAKATioi^r.  1st.  The  first  Solution  is  full. 

2d.  In  the  second  Solution  we  shorten  the  work. 
Finding  that  28  Hundreds  divided 
by  3  give  9  Hundreds,  we  write  9 
Hundreds  in  the  Quotient.  We  then 
multiply  9  Hundreds  by  3 ;  and, 
instead  of  writi7ig  the  Product,  27 
Hundreds,  we  retain  it  in  the  mind, 
and  subtracting  it  from  28  Hun- 
dreds mentally,  find  1  Hundred  re- 
maining. Next,  we  mentally  unite 
the  9  Tens  with  the  1  Hundred  left, 
and  retained  in  the  mind,  and  have 
1  Hundred  and  9  Tens,  or  19  Tens, 
for  a  new  Partial  Dividend,  which 
we  retain  in  the  mind.  Finding 
that  19  Tens  divided  by  3  give  6 
Tens,  we  write  6  Tens  in  the  Quo- 
tient. Multiplying  6  Tens  by  3  and  obtaining  18  Tens 
for  a  Product,  we  subtract  18  Tens  from  19  Tens  men-- 
9 


FIRST  SOLUTION. 

964  Quotient. 
3  Divisor. 

2,892  Dividend. 
27 

19 

18 

12 

12 

SECOND  SOLUTION. 

964 
3 

2,892 


124  FIRST  LESSONS  IN 

tally ;  and,  finding  1  Ten  remaining,  we  retain  this  in 
tlie  mind.  Finally,  uniting  the  2  Ones  of  the  Dividend 
with  the  1  Ten  retained,  we  have  for  our  last  Partial 
Dividend  1  Ten  and  2  Ones,  or  12.  We  do  not  write 
this ;  but,  dividing  by  3  mentally,  we  write  the  result, 
4,  in  the  Quotient.  We  have  964  for  our  final  Quotient. 
The  method  used  in  the  second  Solution  is  much 
shorter  than  that  in  the  first,  since  in  the  second  we 
perform  the  work  mentally ,  without  writing  the  Partial 
Dividends  and  Products, 

DEFINITIONS, 

1,  Iiong  Division  is  the  method  of  Division  used 
where  we  obtain  the  Quotient  by  writing  the  Partial 
Dividends  and  Products, 

2,  Short  Division  is  the  method  of  Division  used 
where  we  obtain  the  Quotient  without  writing  the  Par- 
tial Dividends  and  Products. 

Perform  the  work  by  Short  Division  in  the  following 

Exercises  for  the  Slate  ai^b  Board. 


789,156  -V-  2 

I. 
591,732  ~  2 

769,518  --  2 

768,927  -f-  3 

857,421  -r-  3 

257,613  -f-  3 

736,928  -^  4 

175,628  -^  4 

279,536  --  4 

623,715  -V-  5 

859,235  -^  5 

973,265  -^  5 

157,326  -^  6 

312,714  -f-  6 

517,896  -r-  6 

548,765  -^  7 

227,353  -f-  7 

257,327  -T-  7 

264,976  ^  8 

512,760  -^  8 

512,064  ~  8 

325,719  -4-  9 

289,881  -T-  9 

123,453  -f-  9 

Divide  7,206,480  by 

II. 

2 ;  by  3  ; 

by  4; 

by  5;  by  6. 

Divide  14,405,760 

by 

2;   by3; 

by  4; 

by  5;  by  6. 

Divide  5,040,720 

by 

2;  by3; 

by  4; 

by  5;  by  6. 

Divide   833,280  by 

4;  by5; 

by  6; 

by  7;  by  8. 

MENTAL  AND    WRITTEN  ARITHMETIC, 


125 


LESSON   LXXXIX, 


MUZ  TIP  Lie  A  TION 

TABLE 

DIVISION  TABIES, 

10     1  is 

10, 

10 

10 

1  Time, 

ll 

10 

10  Times, 

10      2  are 

20, 

10 

20 

2  Times, 

2*^ 

20 

10  Times, 

^  10      3  are 

30, 

10 

30 

3  Times, 

3 

30 

10  Times, 

10      4  are 

40, 

10 

40 

4  Times, 

4 

40 

10  Times, 

10  1  0  are 

50, 

10- 

50 

5  Times, 

5.1 

50 

10  Times, 

10 1  6  are 

60, 

10  1 

60 

6  Times, 

6  1 

60 

10  Times, 

10      7  are 

70, 

10 

70 

7  Times, 

7 

70 

10  Times, 

10      8  are 

80, 

10 

80 

8  Times, 

8 

80 

10  Times, 

10      9  are 

90, 

10 

90 

9  Times, 

9 

90 

10  Times, 

10    10  are 

100. 

10 

100  10  Times. 

10 

100 

10  Times. 

Multiplicands. 

10 

I. 
10 

10 

7 

II. 
8        9 

Multijoliers, 

7 

8 

9 

10 

10      10 

Products. 

70 

80 

90 

70 

80       90 

In  these  two  sets  of  Examples,  the  numbers  multiplied 
together  are  the  same,  and  hence  the  Products  must  he 
the  same  in  both  sets. 

From  the  Principle  on  page  93,  we  see  that  if  two 
numbers  are  to  be  multiplied  together  either  may  be  used 
as  the  Multiplicand^  and  the  other  as  the  Multiplier. 
Hence  we  use  the  Multiplicands  of  the  first  set  as  Mul- 
tipliers in  the  second  set. 

The  Products  are  the  same  in  both  sets;  and  are 
formed  in  the  second  set  by  annexing  a  0  to  each  Multi- 
plicand. 

Example.  Multiply  25  by  10. 

Explanation".  Writing  the  10 
under  the  25  so  that  the  1  shall 
stand  under  the  5,  we  bring  down 


25    Multiplicand. 
10  Multiplier. 

250  Product. 


126  FIRST  LESSONS  IN 

the  25  into  the  Product,  and  then  annex  a  cipher,  to 
obtain  the  final  Product.  By  annexing  the  cipher  the 
6  Ones  are  made  5  Tens,  which  are  10  times  as  many  as 
5  Ones ;  and  the  2  Tens  are  made  2  Hundreds,  which 
are  10  times  as  many  as  2  Tens.  Therefore,  since  250 
is  10  times  as  large  as  25,  it  is  the  true  Product.  Hence 
we  have  this 

INFERENCE, 

Annexing  a  cipher  at  the  right  of  a  number  multiplies 
it  hy  10, 

Multiply  each  of  the  following  numbers  by  10  : 
125;     896;    2,570;    37,896;     543,768;     5,020;    500. 


LESSON    XC. 

In  each  of  the  first  4  Equations, 
at  the  right,  2  numbers  are  mul-        tactors.     products. 
tiplied  together  to  make  a  third      2x3         =6 
number.  2x5         =10 

The  numbers  thus  multiplied      3x3  ~  -,  ^ 

together  to  make  a  Product  are      %^\      5  —  qa 
named  Factors.    Factor  means      ^  ^  3  ><  ^  Z  42 
maker  ;  and  Product  means  some- 
thing made,  or  produced. 

Thus,  2  and  5  are  the  Factors  of  10,  their  Product ; 
and  2,  3  and  5  are  the  Factors  of  30,  their  Product. 

The  numbers  6,  10,  9,  15,  30,  and  42,  given  above,  are 
composed  of  the  numbers  multiplied  together  to  make 
them,  and  are  therefore  named  Composite  NuTn- 
bers. 

The  numbers  multiplied  together  to  form  another 
number  are  named  its  Component  Factors. 


MENTAL  AND    WRITTEN  ARITHMETIC.  127 

nEFINITIONS. 

1.  A  Factor  is  one  of  the  numbers  multiplied 
together  to  produce  another  number. 

2,  A  Prime  Factor  is  any  Factor  which  cannot 
itself  be  produced  by  multiplying  together  other  Factors. 

S.  A  Prime  Number  is  any  number  which  cannot 
be  produced  by  multiplying  together  other  numbers. 

4.  A  Composite  Number  is  any  number  which 
can  be  produced  by  multiplying  together  other  numbers. 

5,  A  Com2^07ient  Factor  is  one  of  the  Factors 
multiplied  together  to  produce  a  Composite  IN^umber. 

6,  An  Fven  Number  is  any  number  having  2  as 
one  of  its  Prime  Factors. 

7.  An  Odd  Number  is  any  number  not  having  2 
as  one  of  its  Prime  Factors. 

Eemakks. 

1,  Every  number  whose  right-hand  figure  is  zero,  or 
is  exactly  divisible  by  2,  is  an  Even  Number. 

2.  Every  Even  Number,  except  2,  is  Composite. 

S.  Every  number  whose  right-hand  figure  is  an  Odd 
Number  is  itself  an  Odd  Number. 

Of  the  following  numbers  what  ones  are  Prime,  and 
what  ones  Composite  ?     What  Even,  and  what  Odd  ? 

2,  4,  9,  11,  8,  16,  21,  25,  17,  63,  164. 
6,     7,      6,      10,      13,     14,      15,     19,     22,     27,     120. 

Exercises  for  the  Slate  akd  Board. 

Multiplication. 

Multiply  7,859,634  by  5  ;  by  6  ;  by  7 ;  by  8;  by  9. 
Multiply  9,473,857  by  4 ;  by  6  ;  by  7  ;  by  8 ;  by  9. 
Multiply  5,837,918  by  3  ;     by  5  ;    by  7;    by  8;    by  9. 

Division, 

Divide  6,053,040  by  5  ;  by  6  ;  by  7;  by  8  ;  by  9. 
Divide  6,562,584  by  3  ;      by  4 ;      by  7 ;      by  8 ;     by  9. 


128  FIRST  LESSONS  IN 


LESSON  XGI. 

We  have  heretofore  seen  that  any  Product  can  be 
divided  by  either  of  the  numbers  multiplied  together 
to  produce  it.  In  the  same  manner  any  Product  formed 
of  any  number  of  Factors  can  be  divided  by  any  of  its 
Factors,  or  by  all  of  them  in  succession. 

We  will  take  the  Prime  Factors  13,  7,  and  3,  and  find 
their  Product,  and  then  divide  this  Product  by  its  Prime 
Factors,  13,  7,  and  3,  and  see  what  we  shall  have  for  a 
Quotient. 


\.—Multiplicati(m. 

^.—Division. 

1st  Factor,    13 

13 

2d  Quotient. 

2d  Factor,       7 

_7_ 

2d  Divisor, 

91 

91 

1st  Quotient, 

3d  Factor,       3 

3 

1st  Divisor, 

Product,       273 

273 

1st  Dividend, 

1st,  MuLTiPLiCATioisr.  Multiplying  13  and  7  to- 
gether, and  then  multiplying  their  Product,  91,  by  3y 
the  third  Factor,  we  obtain  273  for  a  final  Product. 

2d,  Divisi02!C.  Dividing  this  Product,  273,  by  3^  the 
last  MultijMer,  we  obtain  91  for  our  first  Quotient. 

DividiJig  91  hy  7^  another  of  the  8  Factors  7nultipUed 
together,  we  obtain,  for  a  Quotient,  13,  the  other  of  the 
8  Factors. 

If,  now,  we  should  divide  our  last  Quotient,  13,  by 
13 f  the  final  Quotient  would  be  1. 

We  have  divided  the  number  273  by  each  of  its  Prime 
Factors  in  succession,  and  obtained  1  for  a  final  Quo- 
tient. Thus  it  is  evident  that,  in  whatever  order  we 
multiply  the  numbers  together,  we  may  divide  the  Pro- 


MENTAL  AND    WRITTEN  ARITHMETIC.  129 

duct  by  all  the  numbers  in  succession,  in  a  contrary 
order* 

Hence  we  have  the  following 

rinciple  in  ZHvision.    A, 

Any  Composite  Number  can  he  divided  ly  any  one  of 
its  Component  Prime  Factors,  or  dy  all  of  them  m  sue- 
cessio7iy  using  each  Quotient  for  a  neio  Dividend. 

Since  any  given  Composite  Number  can  be  formed  by 
multiplying  together  its  Prime  Factor Sy  and  cannot  be 
formed  by  multiplying  together  any  other  Prime  Factors, 
it  is  evident  that  it  cannot  be  exactly  divided  by  any 
other  Prirrie  Factor.    Hence 

Principle  in  Division.    D. 

No  Composite  Number  can  be  exactly  divided  by  any 
Prime  Number  which  is  not  one  of  its  Prime  Factors. 
Hence, 

To  FIND  THE  Prime  Factors  of  any  Number: 

EULE. 

Divide  the  Number  by  any  Prime  Number  that  tvill 
exactly  divide  it ;  then  divide  the  Quotient  by  any  Prime 
Number  that  will  exactly  divide  it ;  and  so  continue  to 
divide  until  a  Quotient  is  obtained  that  is  itself  a  Prime 
Number.  The  several  Divisors  and  the  final  Quotient 
will  be  the  Prime  Factors  of  the  given  Prime  Number. 

Find  and  write  all  the  Prime  Factors  of  each  of  the 
following  Composite  ISTumbers : 


16 

24 

30 

40 

50 

60 

75 

85 

100 

18 

25 

32 

42 

54 

64 

80 

90 

112 

20 

27 

35 

45 

55 

70 

81 

95 

120 

21 

28 

36 

48 

56 

72 

84 

96 

128 

130 


FIRST  LESSONS  IN 


365  Multiplicand. 
9  1st  Multiplier. 

3,285 

2  M  Multiplier. 

6,570  Product. 


LESSON   XGII. 

Example,  Multiply  365  by  18. 

ExPLAKATioiq-.  1st.  We  separate  the  Multiplier,  18, 
into  the  Component  Factors  2  and  9. 

2d.  We  multiply  365  by  9,  one 
of  the  Component  Factors  of  18, 
thus  taking  365   9  times, 

3d.  We  multiply  this  Product, 
3,285,  by  2,  the  other  Component 
Factor  of  18 ;  thus  taking  9  times 
365  2  times.  Since  2  times  9 
times  are  18  times,  we  have  thus  taken  365  18  times ; 
or  multiplied  365  by  18. 

Instead  of  2  and  9,  as  the  Component  Factors  of  18,  we 
might  have  taken  3  and  6,  or  2,  3  and  3.  Multiplying  by 
these,  we  should  have  obtained  the  same  Product.   Hence, 

To  Multiply  by  any  Composite  Number: 

EULE. 

I.  Separate  the  Multiplier  into  any  number  of  Com^ 
ponent  Factors. 

II.  Multiply  the  Multiplicand  ly  one  of  these  Factors  ; 
then  this  Product  hy  another,  and  so  on  till  all  the  Com- 
ponent Factors  have  leen  used  as  Multipliers.  The  final 
Product  will  be  the  true  Product. 

By  using  Component  Factors  as  Multipliers, 
Multiply  23,546  by  12 
Multiply  37,985  by  24 
Multiply  85,896  by  42 
Multiply  98,583  by  56 
Multiply  56,874  by  75 


by  14 

by  15 

by  18. 

by  27 

by  35 

;    by  36. 

by  45^ 

by  48 

by  49. 

by  63; 

by  64 

by  70. 

by  81, 

by  84^ 

by  96. 

MENTAL  AND    WRITTEN  ARITHMETIC.  131 

LESSON   XGIIL 

^iriDIJVG    ^r  A    COM'POSITB  jyUMl^B^. 

l.—MuUiplication.  2.— Division. 

Multiplicand,      376  376  Final  Quotient. 

1st  Multiplier. 7  7  2d  Divisor. 

1st  Product.     2,632  2,632  1st  Quotient. 

2d  Multiplier.    2  2  1st  Divisor. 

Final  Product.  5,264  5,264  Dividend. 

Above  we  have  multiplied  376  by  14,  by  using  7  and 
2,  the  Component  Factors  of  14,  as  Multipliers.  The 
Product  is  5,264.  If,  now,  this  Product  be  divided  by 
14,  our  Multiplier,  the  Quotient  will  be  the  Multipli- 
cand, 376.  But,  above,  we  have  obtained  this  result  by 
dividing  by  2  and  7,  the  Component  Factors  of  14. 

We  notice  that  in  our  Division  we  retraced  all  our 
work  of  Multiplication.    Hence, 

To  Divide  by  ant  Composite  Number: 

EULE. 

I.  Separate  the  Divisor  into  any  number  of  Component 
Factors. 

II.  Divide  the  Dividend  ly  any  one  of  these  Factors  ; 
then  divide  the  Quotient  by  another  Factor,  and  so  on 
till  all  the  Factors  have  been  used  as  Divisors. 

The  final  Quotient  will  be  the  true  Quotient. 
By  using  Component  Factors  as  Divisors, 


Divide      40,320  by  12 ; 

by  14; 

by  15; 

by  16. 

Divide     816,480  by  24 ; 

by  27; 

by  35; 

by  36. 

Divide  4,445,280  by  42 ; 

by  45; 

by  48; 

by  49. 

The  figures  1,  2,  3,  4,  5,  6,  7,  8,  9,  always 
Member.  They  are  therefore  called  Numeral  Fig' 
tireSf  or  Nuinerals.  Zero  (0)  never  itself  _  expresses 
Number,  but  shows  the  absence  of  Number. 


132  FIRST  LESSONS  IN 

LESSON   XCIV. 

SISTIJ^G     OJ^    /     WITH     CI^H^^S    A.J\r- 

Example.  Multiply  365  by  100. 

ExpLAKATio:^-.  10  and  10  may  be  regarded  as  the 
Component  Factors  of  100.  Hence  we  can  multiply  by 
100  by  multiplying  by  10  and  10.  But  we  may  multi- 
ply by  10  by  annexing  one  cipher  at  the  right  of  the 
Multiplicand.  Hence,  to  multiply  by  10  twice,  that  is, 
by  100,  we  must  annex  tico  ciphers.  Therefore,  365  x 
100  =  36,500.  The  ciphers  at  the  right  of  1,  in  100, 
or  1,000,  or  10,000,  or  any  other  number,  show  how 
many  times  10  is  a  factor  in  the  number.    Hence, 

To  Multiply  by  any  Number  consisting  of  1 
WITH  Ciphers  annexed  at  the  right: 

EULE. 

Annex  at  the  right  of  the  Multiplicand  as  many  ci- 
phers as  there  are  in  the  Multiplier,     TJie  result  will  he 
the  true  Product, 
Multiply  785  by  1,000 ;  by  10,000 ;         by  100,000. 

Multiplying  by  any  Number  consisting  of  one 
Numeral  Fig  ure  with  Ciphers  annexed  : 

The  Component  Factors  of  500  may  be  taken  as  5  and 
100;  since  these  numbers,  multiplied  together,  make 
500.  Hence,  we  can  multiply  by  500  by  multiplying  by 
its  Factors,  5  and  100.  It  is  also  plain  that  we  can  mul- 
tiply by  5,000  by  multiplying  by  its  Factors,  5  and 
1,000.  Therefore,  to  multiply  by  5,000,  we  can  first 
multiply  by  5,  and  then  multiply  the  Product  thus  ob- 
tained by  1,000,  by  annexing  three  ciphers.     Hence, 


MENTAL  AND  WRITTEN  ABITH:^ETIC,  133 

To  Multiply  by  a  Number  consisting  of  a  sin- 
gle Numeral  Fig  ure,  with  Ciphers  annexed  : 

KULE. 

I.  MuUijjly  the  Multiplicand  hy  the  left-hand  figure 
of  the  Multi^jlier. 

II.  Annex  at  the  right  of  this  Product  as  many 
Ciphers  as  there  are  standing  at  the  right  in  the  Multi- 
plier,    Tlie  result  will  be  the  true  Product. 

Exercises  for  the  Slate  ai^d  Board. 
7,854  X  70  3,586  x  700  734  x  5,000 

9,873  X  90-  8,962  x  800  548  x  7,000 

5,420  X  80  5,423  x  900  820  x  9,000 

In  the  number  555,  the  figures  do  not  each  express 
the  same  vahie.  The  5  at  the  right  stands  for  5  Ones ; 
the  5  at  the  left  of  this  for  5  Tens,  or  50 ;  and  the  5  at 
the  left  for  5  Hundreds,  or  500.  Hence,  the  value  of 
each  figure  5  is  determined  by  its  place  or  locality. 

DEFINITIONS. 

1.  The  Simple  Value  of  any  Numeral  Figure  is 
the  value  which  it  expresses  when  standing  in  the  2^l(^oe 
of  Ones. 

2.  The  Local  Value  of  any  Numeral  Figure  is  the 
value  which  it  expresses  when  standing  in  any  place. 

Eemarks. 

1.  The  Simple .  Value  of  a  Numeral  Figure  is  always 
the  same. 

2.  The  Local  Value  of  a  Numeral  Figure  changes  as 
often  as  the  place  of  the  figure  is  changed. 

3.  When  a  Numeral  Figure  stands  in  the  place  of 
Ones,  its  Simple  and  Local  Value  are  the  same. 

4.  The  figure  0  has  no  value,  either  Simple  or  Local. 


134 


MBST  LESSONS  IN 


LESSON   XCIV. 

Example  A.     Multiply       solution. 
549  by  375. 

Explanation.  1st.  We 
multiply  549  by  5,  or  take  2,745 
it  5  times,  and  write  the  38,430 
Partial  Product,  2,745.  2d.  ^^^^^^^ 
We  multiply  549  by  70,  or  205,875 
take  it  70  times,  and  write 

the  Partial  Product,  38,430.  3d.  We  multiply  549  by 
300,  or  take  it  300  times,  and  write  the  Partial  Product, 
164,700.  4th.  We  add  the  3  Partial  Products,  and  take 
their  Sum  as  the  final  Product. 

Thus  we  have  taken  549,  our  Multiplicand,  300  times, 
and  70  times,  and  5  times,  or  875  times.  Hence,  we 
have  the  true  Product. 


549  Multiplicand. 
375  Multiplier, 

5  Times  5Jf9. 

70  Times  BJfi. 

800  Times  5Jfi, 

875  Times  5J^9. 


249 
_J05 

1,245 

_74^00 

75,945 


Example  B.  Multiply  249  by  305. 

Explanation.  We  first  take  249  5  times, 
then  800  times,  and,  writing  the  Partial 
Products,  add  them. 

We  do  not  have  any  Partial  Product 
arising  from  multiplying  by  0  in  the  Mul- 
tiplier ;  since  taking  249  no  (0)  times  is  not  taking  it  at 
all.  The  other  Partial  Products  are  written  as  in  the 
preceding  Solution. 

For  each  Numeral  Figure  of  the  Multiplier  there  is  a 
corresponding  Partial  Product,  obtained  by  multiplying 
the  whole  Multiplicand  by  this  Figure.     Hence, 

To  Multiply  one  Number  by  another: 

EULE. 

I.   Write  the  Multiplier  under  the  Multiplicands 


MENTAL  AND    WRITTEN  ARITHMETIC. 


135 


II.  Commencing  at  the  right,  midtij^ly  the  whole  MuU 
tijMcand  hy  each  Numeral  Figure  of  the  Multiplier, 
regarding  the  Local  Value  of  each  Figure,  and  write  the 
Partial  Products, 

III.  Add  the  Partial  Products,  and  tahe  their  Sum  as 
the  final  Product. 

EXEKCISES  FOR  THE   SlATE  AKD   BoARD. 

Multiply  3,582  by  125  ;  by  342 ;  by  976  ;  by  748. 
Multiply  7,643  by  502 ;  by  430 ;  by  900 ;  by  708. 
Multiply  9,536  by  739 ;      by  608 ;      by  968 ;      by  374. 

Multiply  5,894  by  5,423  ;  by  4,672 ;  by  6,702 ;  by  3,064. 
Multiply  3,698  by  5,008 ;  by  7,041 ;  by  8,302 ;  by  7,006. 

Multiply  7,874  by  25,376  ;  by  30,708 ;  by  50,023. 
Multiply     657  by  46,002 ;         by  50,007 ;        by  10,101. 


LESSON   XCV. 


Example.  Multiply  347  by 
235. 

ExPLAKATiON".  The  first  of 
these  two  Solutions  is  in  the 
form  heretofore  used. 

The  second  Solution  differs 
from  the  first  only  in  having 
ciphers  omitted  at  the  right  of 
the  Partial  Products. 

We  observe  that  the  second  Partial  Product,  10,41,  is 
found  by  multiplying  347  by  3  ;  and  the  third,  69,4,  by 
multiplying  347  by  2.  But,  since  the  figure  3  in  the 
Multiplier  stood  for  30,  or  3  x  10,  10,41  must  still  be 


1st  solution. 

347 
_235 

1,735 
10,410 
69,400 

81,545 


2d  solution. 

347 
235 

1,735 

10,41 
69,4 

81,545 


136  FIRST  LESSONS  IN 

multiplied  by  10.  It  has  been  written  one  place  to  the 
lefty  so  as  to  leave  room  for  a  cipher  at  the  right ,  on  mul- 
tiplying it  by  10.  So,  also,  the  Partial  Product  69,4  has 
been  written  two  places  to  the  left,  so  as  to  leave  room 
for  two  ciphers  at  the  right,  on  multiplying  by  100. 

In  the  second  Solution  the  right-hand  figure  of  each 
Partial  Product  is  written  directly  helow  the  correspond- 
ing figure  of  the  Multiplier,  Hence,  when  the  ciphers 
at  the  right  of  the  Partial  Products  are  omitted, 

To  Write  the  Partial  Products: 

EULE. 

Multiply  the  Multiplicand  hy  each  Numeral  Figure  of 
the  Multiplier,  and  write  the  Partial  Products  so  that 
the  EIGHT-HAND  FIGURE  in  each  shall  stand  directly 

BELOW  THE  CORRESPOKDIKG  FIGURE  OF  THE  MULTI- 
PLIER. 

If  there  are  any  ciphers  at  the  right  of  the  first  Nu- 
meral Figure  of  the  Multiplier,  write  the  same  number 
of  ciphers  at  the  right  of  the  first  Partial  Product 

The  second  Partial  Product,  10,41,  is  read  10,410; 
and  the  third,  69,4,  is  read  69,400.    Hence, 

To  Bead  the  Partial  Products  : 

EULE. 

Read  each  Partial  Product  the  same  as  if  the  omit- 
ted CIPHERS  WERE  WRITTEN. 

Exercises  for  the  Slate  and  Board. 

Multiply  6,852  by  .237  ;  by  543  ;      by  678  ;       by  906. 

Multiply  3,826  by  460 ;  by  307  ;      by  500;    by  1,060, 

Multiply  6,978  by  1,502 ;  by  2,007;  by  2,030;  by  9,706. 

Multiply  8,059  by  6,025;  by  3,006;  by  9,408;  by  3,500. 

Multiply  7,684  by  2,643 ;  by  5,489;  by  3,762;  by  7,563. 


IIENTAL  AND    WRITTEN  ARITHMETIC,  137 


LESSON   XGVI, 

Example.  Find  the  Product  arising  from  multiply- 
ing together  468  and  12. 


FIRST  SOLUTION. 

12  MiiUiplicand. 
468  MuUiplier. 

96  =      8  ti7nes  12. 

72    =    60  times  12. 

4,8      =  iOO  times  12. 

5^  =  ^68  times.  12. 


SECOND  SOLUTION. 

4jS8  Multiplicand, 
12  Multiplier. 

96  =  12  times     8. 

72    =  12  times    60. 

4,8_  =  12  times  JfiO. 

5,616  =.  12  times  1^68. 


ExPLAN'ATiOi^r.  As  the  Product  will  be  the  same 
whichever  of  the  two  numbers  be  taken  as  Multiplier, 
in  the  first  Solution  we  have  made  468  Multiplier,  and 
in  the  second  Solution  12.  The  first  Solution  is  in  the 
usual  form. 

In  the  second  Solution  we  first  multiplied  8  by  12; 
then  6,  or  60,  by  12;  and  finally  4,  or  400,  by  12.  Thus 
we  have  taken  400,  and  60,  and  8,  12  times ;  or  have 
taken  468  12  times.  The  Partial  Products  stand  in  tlie 
same  order,  and  are  the  same,  in  hotli  Solutions. 

In  the  second  Solution,  we  find  it  easier  to  obtain  the 
Product  of  8  multiplied  by  12  by  multiplying  12  hy  8, 
the  Product  being  the  same  in  both  cases.  So,  also,  to 
find  the  Product  of  60  multiplied  by  12,  we  multiply  12 
by  60.  In  the  same  manner,  we  find  the  Product  of  400 
X  12  by  multiplying  12  by  400. 

The  peculiarity  in  the  second  Solution  consists  in 
multiplying  first  the  right-hand  figure  of  the  Multipli- 
cand hy  the  entire  Multiplier,  then  the  next  figure  of  the 
Multiplica7id  hy  the  entire  Multiplier,  and  so  on  until 
all  the  figures  of  the  Multiplicand  have  been  multiplied 
by  the  Multiplier.     But,  though  we  consider  X'^  the 


138 


FIRST  LESSONS  IN 


Multiplier,  we  obtain  the  Partial  Products  by  using  the 
figures  of  the  Multiplicand  as  Multipliers,  and  12  as 
Multiplicand. 

By  the  method  used  in  the  second  Solution,  obtain 
the  Products  in  the  following 

EXEECISES   FOR  THE   SlATE   AND   BOARD. 

Multiply    473  by  13;  by  25  ;    by  36  ;    by  47;    by  58. 

Multiply     568  by  45;  by  73  ;    by  84;    by  39  ;    by  96. 

Multiply  2,437  by  123 ;  by  234 ;      by  543 ;      by  736. 

Multiply  5,283  by  542 ;  by  745 ;      by  827 ;      by  976. 


LESSON   XCVIL 

Example  A.  Multiply  236  by  12. 
Example  B.  Divide  2,832  by  12. 


Solution  op  Ex.  A. 

Multiplicand.         236 
MultijMer.  12 

12  times      6  =     ^2 
12  times    SO  =     36 
12  times  200  =_2^_ 

2^32 


Solution  op  Ex.  B. 

236  Quotient, 
12  Divisor. 

^,832  Dividend. 

2,4       1=  12  times  200. 


43 

36_  =  12  times 

72 

72  =  12  times 


6. 


Expla:n'ATIOIT.  In  the  Solution  of  Example  A,  the 
Product  is  obtained  by  the  method  used  in  the  last 
Lesson.  We  find  the  Product  to  be  2,832  ;  the  same  as 
the  Dividend  in  Example  B.  Our  Multiplicand  is  236, 
our  Multiplier  12,  and  our  Product  2,832. 

In  Example  B  we  are  required  to  divide  this  Product, 
2,832,  by  the  Multiplier,  12.    Hence,  according  to  Prin- 


MENTAL  AND    WRITTEN  ARITHMETIC.  139 

ciple  5  in  Diyision,  on  page  106,  our  Quotient  must  be 
the  same  as  our  Multiplicand,  236.  We  will  obtain  it 
by  the  Solution. 

Writing  the  Divisor  above  the  Dividend,  with  a  line 
between  them,  we  see  that  we  are  to  find  a  Multiplicand 
which  when  multiplied  by  12  will  give  2,882  for  a 
Product. 

1st.  We  first  ask:  What  is  the  greatest  number  of 
Hundreds  which,  when  written  in  the  Multiplicand  (or 
Quotient)  and  multiplied  by  12,  will  give  a  Partial 
Product  not  exceeding  28  Hundreds  ?  Finding  this 
number  of  Hundreds  to  be  2,  we  write  2  Hundreds  in 
the  Multiplicand  (or  Quotient).  Multiplying  2  Hundreds 
by  12,  or  12  by  200,  and  subtracting  the  Partial  Product, 
24  Hundreds,  from  28  Hundreds,  4  Hundreds  are  left. 

2d.  Bringing  down  the  3  Tens  from  2,832,  and  writ- 
ing them  at  the  right  of  4  Hundreds,  we  have  4  Hun- 
dreds and  3  Tens,  or  43  Tens,  for  our  next  Partial  Divi- 
dend. We  now  ask :  What  is  the  greatest  number  of 
Tens  which,  when  written  in  the  Multiplicand  and 
multiplied  by  12,  will  give  a  Partial  Product  not  exceed- 
ing 43  Tens.  Finding  this  number  of  Tens  to  be  3,  we 
write  3  Tens  in  the  Multiplicand  (or  Quotient),  and 
multiplying  the  3  Tens  by  12  (or  12  by  30),  and  sub- 
tracting the  Partial  Product,  36  Tens,  from  43  Tens,  we 
write  the  Eemainder,  7  Tens. 

3d.  Finally,  bringing  down  the  2  Ones  from  2,832, 
and  writing  them  at  the  right  of  our  7  Tens,  we  have  7 
Tens  and  2  Ones,  or  72  Ones,  for  a  Partial  Dividend. 
Finding  that  6  Ones,  when  written  in  the  Multiplicand 
(or  Quotient)  and  multiplied  by  12,  will  give  72  Ones, 
we  write  6  Ones  as  the  last  figure  in  the  MultipHcand 
(or  Quotient).  Multiplying  and  subtracting,  as  before, 
10 


140  FIRST  LESSONS  IN 

we  have  no  Remainder.   Therefore^  our  Multiplicand,  or 
Quotient,  is  236.    Hence, 

To  Divide  one  Number  by  another  : 

EULE. 

I.  Write  the  Divisor  above  the  Dividend,  separating 
them  ly  a  horizontal  line, 

II.  Commencing  at  the  left,  take  as  a  Partial  Dividend 
such  a  part  of  the  entire  Dividend  as,  without  regarding 
its  Local  Value,  ivill  contain  the  Divisor  at  least  okce^ 
and  NOT  MOKE  than  nine  times  ;  and,  determining  the 
first  figure  of  the  Quotient ^  write  it  in  its  place,  giving  it 
its  proper  Local  Value, 

III.  Multip)ly  this  Divisor  ly  the  Quotient  figure,  and 
subtract  the  result  from  the  Partial  Dividend, 

IV.  Write  the  next  figure  of  the  Dividend  at  the  right 
of  the  Remainder ;  and,  using  the  number  thus  formed 
as  a  new  Partial  Dividend,  determine  the  second  figure 
of  the  Quotient  in  the  same  manner  as  the  first  was  ob- 
tained;  and  multiply  and  subtract  as  before. 

Proceed  in  this  manner  till  all  the  figures  of  the  Quo- 
tient are  obtained,  and  the  worh  is  completed. 

Kemarks. 

1.  If  any  Partial  Product  is  greater  than  the  Partial 
Dividend  under  which  it  is  written,  the  corresponding 
Quotient  figure  is  too  large,  and  must  be  diminished. 

2.  If  any  Remainder  is  greater  than  the  Divisor,  the 
corresponding  Quotient  figure  is  too  small,  and  must  be 
increased. 

3.  Wheneyer  any  Partial  Dividend  is  less  than  the 
Divisor,  the  corresponding  Quotient  figure  is  0 ;  and 
the  next  Partial  Dividend  is  formed  directly  from  this, 
by  writing  the  next  figure  of  the  Dividend  at  the  right 
of  it. 


3£ENTAL  AND    WRITTEN  ARITHMETIC.  141 

EXEECISES  FOR  THE   SlATE  AND   BOARD. 


Divide 

33,760 

by  12; 

by  13; 

by  14; 

by  15. 

Divide  1,970,640 

by  23; 

by  34; 

by  45; 

by  56. 

Divide 

375,480 

by  149 ; 

by  398 ; 

by  447 ; 

by  745. 

Divide 

134,488 

by  347; 

by  494 ; 

by  741 ; 

by  1,739. 

LESSOlV  XCVIII. 

Example.  Divide  1,589  by  58. 

ExPLA:NrATiO]sr.  We  divide  as  here-  solution. 

tofore,  and  obtain  27  for  a  Quotient;  ^^  Quotient. 

but,  on  subtracting  the  last  Partial      ^  Divisor. 

Product,  we   have   23   for   a   Re-  1,^^^  Dividend. 

mainder.    This  last  Remainder  is  ^^^^ 

named  the  Final  Remainder.  429 

It  is  never  called  the  Difference,  as  _^^ 

in  Subtraction.  23  Remainder. 

DEFINITION. 

The  Final  Hemainder  in  Division  is  that  part 
of  the  Dividend  left,  still  undivided,  after  obtaining  all 
the  figures  of  the  Quotient. 

Exercises  for  the  Slate  akd  Board. 

Divide        23,578  by  35  ;      by  59  ;      by  68  ;      by  79. 
Divide       376,982  by  143  ;     by  256  ;    by  374 ;    by  578. 
Divide      438,796  by  527 ;    by  743  ;    by  963  ;    by  829. 
Divide  25,762,159  by  159  ;  by  1,524 ;  by  3,284. 

Divide  57,349,284  by  6,023  ;       by  7,009  ;  by  8,219. 

Divide  29,513,784  by  374 ;  by  183  ;  by  921 ;  by  548. 
Divide  73,182,546  by  1,316  ;  by  589  ;  by  918  ;  by  3,600. 
Divide  10,020,010  by  101 ;  by  1,010 ;  by  10 ;  by  100. 
Divide  10,000,000  by  10;  by  100;  by  1,000;  by  10,000. 


14a  FIRST  LESSONS  IN 

LESSON   XO/X. 

We  multiply  by  10,  100,  1,000,  &c.,  by  annexiDg  at 
the  right  of  the  Multiplicand  as  many  ciphers  as  stand 
at  the  right  in  the  Multiplier. 

According  to  Principle  5,  on  page  106,  if  the  Product 
be  divided  by  the  Multiplier  the  Quotient  will  be  the 
Multiplicand.  But  it  is  evident  that  we  can  obtain  the 
Multiplicand,  or  Quotient,  in  such  cases,  by  dropping  at 
the  right  of  the  Product,  or  Dividend,  the  same  ciphers 
which  we  have  just  annexed.    Hence, 

To  DIVIDE  BY  10,  100,  1,000,  &C.,  WHEN  THE  DIVI- 
DEND HAS  AS  MANY  CIPHERS  AT  THE  RIGHT  AS 
ARE  FOUND  IN  THE  DiVISOR : 

KULE. 

Drop  at  the  right  of  the  Dividend  as  many  ciphers  as 
stand  at  the  right  in  the  Divisor,  The  figures  remaining 
will  he  the  Quotient. 

Exercises  eor  the  Slate  and  Board. 
Divide  57,830,000  by  100 ;        by  1,000  ;        by  10,000. 
Divide  21,700,000  by  10 ;  by  10,000  ;      by  100,000. 

Divide  65,430,000  by  10 ;  by  1,000 ;        by  10,000. 

Example  1.  Divide  54,768  by  100. 

EXPLAKATION^  1.   Before  solution. 

dividing,  we  separate  our    54,700  4-68. 

Dividend  into  tico  parts,    54^^100=547  Quotient. 

,54,700  and  68.     l^ext  we    Fi7ial  Rem.,       68. 

divide  the  first  part,  54,700 

by  100,  by  dropping  two  ciphers,  and  have  547  for  a 

Quotient.     It  is  plain  that  68  will  not  contain  100  even 

once.    Hence  it  will  be  our  Final  Eemainder.     Our 

Quotient  is  547,  and  our  Final  Eemainder  68. 


MENTAL  AND    WRITTEN  ARITHMETIC, 


143 


54,768  =  54,000  +  768 
54,000 +-1,000=54  QuoH. 
Final  Remainder,  768. 


Example  2.  Divide  54,768 
by  1,000. 

ExPLAis^ATiOK.  Eemoving 
the  three  right-hand  figures, 
768,    and    supplying    their 

places  with  ciphers,  we  separate  the  Dividend  into  two 
parts,  54,000  and  758.  Dividing  54,000  by  1,000,  by 
rejecting  the  three  ciphers  at  the  right,  we  obtain  54  for 
our  Quotient.  The  other  part  of  the  Dividend,  768, 
being  less  than  the  Divisor,  will  not  contain  it  even 
once,  and  hence  will  be  our  Final  Kemainder.  • 

On  examination,  we  find  that  when  we  divide  by  100, 
1,000,  &c.,  the  figures  removed  at  the  right  of  the  Divi- 
dend form  our  Final  Kemainder,  and  the  figures  at  the 
left  of  these,  in  the  Dividend,  form  our  Quotient.  Hence, 

To  Divide  by  10,  100,  1,000,  &c.: 

EULE. 

Remove  at  the  right  of  the  Dividend  as  many  figures 
as  there  are  ciphers  in  the  Divisor,  and  take  the  number 
composed  of  the  figures  so  removed  for  the  Final  Re- 
mainder, and  the  number  composed  of  the  figures  at  the 
left  of  these  for  the  Quotient. 


Example  3.  Di- 
vide 7,958  by  400. 

Explanation". 
The  first  Solution  is 
in  the  form  hereto- 
fore given.  We  see 
that  the  ciphers  in 
the  Divisor  appear 
in  the  Partial  Prod- 
ucts, but  do  not  affect  the  work  so  as  to  change  either 
the  Quotient  or  the  Final  Remainder. 


1st  solution. 

19       QuoH. 
_400  DivW, 

7958  Divided, 
400_ 

3958 
3600 

358  Fin.  Rem, 


2d  solutiok 

19        QuoH. 

4 1  GO  Div'r. 

79|58  Divided. 
4_ 

39 
36_ 

358   Fin.  Rem. 


144 


FIRST  LESSONS  IN 


The  figures  58,  at  the  right  in  the  Dividend,  appear 
in  the  Final  Eemainder  unchanged.  Hence,  in  the  sec- 
ond Solution  the  work  is  shortened,  by  cutting  off  the 
ciphers  at  the  right  in  the  Divisor,  and  the  figures  58  at 
the  right  in  the  Dividend,  and  then  annexing  the  58  to 
the  second  Eemainder.     Therefore, 

To  Divide  by  a  Number  with  Ciphers  at  the 
Right: 

KULE. 

I.  Cut  off  the  ciphers  at  the  right  of  the  Divisor,  and  an 
equal  rjLumber  of  figures  at  the  right  of  the  Dividend, 

II.  Divide  the  Dividend,  thus  changed,  by  the  changed 
Divisor,  and  use  the  Quotient  thus  obtained  as  the  true 
Quotient. 

III.  Annex  to  the  last  Eemainder  the  figures  cut  off  from 
the  Dividend,  and  use  the  number  thus  formed  as  the  Final 
Remainder, 

EXEKCISES  FOR  THE   SlATE  Al^D  BOAKD. 


Divide    78,546  by    10 

;           by     100 

;           by    1,000. 

Divide    57,968  by  100 

;           by  1,000 

;            by  10,000. 

Divide    30,102  by    10^ 

by     100 , 

by    1,000. 

Divide    73,001  by  100 

;           by  1,000 

;           by  10,000. 

Divide  237,849  by    60^ 

by     700 

by    8,000. 

Divide  546,789  by    80 

by     900 , 

by  70,000. 

Divide  107,050  by    90; 

by     500; 

by    8,000. 

Divide  700,520  by    30^ 

by     650 

by    3,500. 

Divide  672,518  by  230  ^ 

by  3,100; 

by  30,100. 

Divide  127,950  by  100 

by  7,200 ; 

by  50,001. 

Divide  718,312  by  101 

by  2,001 ; 

by  10,001. 

MENTAL  AND    WRITTEN  ARITHMETIC.  145 

LESSON   C. 

Example.  ist  solution.       2d  solution. 

Multiply  15  by      Multiplicand,  15  =  3x5 

14  Multiplier.       14=  2x7 

ExPLAiq-A-  60     3x5x2x7 

Tioisr.  1st.  The  1^  _^ 

first   Solution      Product.        210  14 

is  in  the  usual  __^ 

form,  and  the  Product  is  210.  2d.  In  the  70 

second  Solution,  we  factor  15  into  3  and        3 

5,  and  14  into  2  and  7.     Writing  the        210  FrodH, 
Factors  of  both  Multiplicand  and  Mul- 
tiplier in  a  line,  with  the  Sign  x  between,  for  the  Fac- 
tors of  the  Product,  we  multiply  them  together,  and 
obtain  210  for  a  Product.     Hence, 

General  Principle  in  Multiplication. 

The  Product  is  composed  of  the  Factors  of  the  Multipli- 
cand and  Multiplier,  and  IS'O  othehs. 

Since  the  Product  in  Multiplication  becomes  the  Divi- 
dend in  Division,  the  Multiplier  the  Divisor^  and  the 
Multiplicand  the  Quotient,  therefore,  in  Division,  when 
there  is  no  Remainder,  we  have 

General  Principles  in  Division. 

1.  The  Dividend  is  composed  of  the  Fdctors  of  the  Divi- 
sor and  Quotient,  and  KG  others. 

2.  If  the  Factors  of  the  Divisor  he  rejected  from  the 
Dividend,  the  remaining  Factors  will  be  those  of  the  Quo- 
tient.   Hence, 

To  Multiply  by  any  Number: 

EULE. 

Connect  the  Factors  of  the  Multiplier  with  those  of  the 
Multiplicand  by  the  Sign  x .  The  Product  of  these  Fac- 
tors will  be  the  true  Product 

7 


146  FIRST  LESSONS  IN 

Hence,  also,  when  there  is  no  Remainder^ 
To  Divide  by  any  Number: 

KULE. 

From  the  Dividend  remove  Factors  equal  to  those  of  the 
Divisor,  The  Product  of  tJie  remaining  Factors  will  he 
the  true  Quotient, 

We  saw,  on  page  98,  that  the  Sign  of  Division,  -^,  is 
made  from  the  Sign  Minus,  — . 

We  may  use  the  Sign  Minus  in  still  another  manner 
to  show  that  one  number  is  to  be  divided  by  another. 
We  may  show  that  12  are  to  be  divided  hy  3,  in  the 
manner  seen  at  the  right. 

1st :  We  write  the  Divi-    Dividend,  12  _ 
dend,  12.     2d:   We  write    Divisor,      3  "  ^  V^^^^^^^- 
the  Sign  Minus  below  the 

Dividend.    3d:  We  write  the  Divisor  below  the  Sign 
Minus. 

If  we  make  this  Expression  the  First  Member  of  an 

Equation,  the  Second  Member  will  be  the  Quotient. 

12 
The  Expression  —  =  4  is  read :  "  12  divided  ly  3  equal 

^"    The  Quotient,  4,  is  called  the  Value  of  the  Ex- 

.      12 
pression  — 

Find  the  Value  of  each  Expression  in  the  following 

Exercises  for  the  Slate  ai^d  Board. 
I. 


15  =  ? 

5 

27  _ 
3 

? 

43  _ 

7  ~ 

11. 

? 

72  _p 
8 

63  _p 

7 

936 

675 

959 

1792 

3753 

3 

5 

7 

8 

9 

MENTAL  AND    WRITTEN  ARITHMETIC.  147 

Example.  Divide  378  by  42. 

ExPLAi^ATiON.  From  the  second  General  Principle  in 
Division  we  see  that  the  Quotient  consists  of  those  Fac- 
tors of  the  Dividend  which  are  not  in  the  Divisor. 
Hence,  by  the  method 

shown  on  page  129  we  solution. 

factor  378  into  2x3      378=2  x  3  x  7  x  9. 
X  7  X  9,  and  42  into        42=2  x  3  x  7.    Hence, 
2x3x7.     Writing      378^2  x3  x  7  x_9^^  Quotient. 
the  Factors  of  the  Di-       42     2x3x7 
vidend  above  those  of 

the  Divisor,  for  Division,  we  reject  from  the  Dividend 
the  Factors  2  x  3  x  7,  of  the  Divisor,  as  directed  by 
the  preceding  Rule,  and  have  the  remaining  Factor,  9, 
for  our  Quotient. 

Obtain  the  Quotients  by  factoring  in  the  following 

EXEECISES  FOR  THE   SlATE  Al^D   BOARD. 

132  195  2205  2940  3888  864j0 
66     39     441     420     432     1728 


LESSON    01. 


Since  our  Quotient  always  consists  of  those  Factors  of 
the  Dividend  remaining  after  taking  away  the  Factors 
of  the  Divisor,  it  is  plain  that  (Principle  3)  putting  any 
neiu  Factor  into  the  Dividend  does  in  effect  put  that 
Factor  into  the  Quotient ;  and  that  (Principle  4)  remov- 
ing from  the  Dividend  any  Factor  already  there  in  effect 
removes  it  from  the  Quotient, 

It  is  also  evident  that  (Principle  5)  if  any  new  Factor 
be  put  into  the  Divisor,  when  we  take  the  Factors  of  the 


148  FIRST  LESSONS  IN 

Divisor  from  the  Dividend  we  must  also  take  this  new 
Factor  from  the  Dividend^  and  thus  in  effect  take  it  from 
the  Quotient,  or  divide  the  Quotient  by  it;  and  that 
(Principle  6)  if  any  one  of  the  Factors  of  the  Divisor 
be  removed  from  the  Divisor  before  taking  the  Factors 
of  the  Divisor  from  the  Dividend,  the  Factor  so  removed 
will  not  be  taken  from  the  Dividend,  as  it  should  be, 
but  will  leave  the  corresponding  Factor  of  the  Dividend 
in  the  Quotient,  and  will  thus,  in  effect,  multiply  the 
Quotient  by  this  Factor. 

It  is  also  clear  that  (Principle  7)  if  any  new  Factor 
be  put  into  both  Dividend  and  Divisor,  the  one  in  the 
Divisor  will  cause  the  removal  of  the  one  in  the  Divi- 
dend, and  hence  the  Quotient  will  not  be  affected.  And 
(Principle  8)  if  the  same  Factor  be  removed  from  both 
the  Dividend  and  Divisor,  the  final  effect  will  be  the 
same  as  if  it  had  remained  in  both  till  all  the  Factors 
of  the  Divisor  were  taken  from  the  Dividend.  Hence 
the  Quotient  will  not  be  affected. 

Therefore,  we  shall  have  as 

General  Frinciples  of  Division  : 

3.  Multiplying  tJie  Dividend  hy  any  Factor  in  effect 
multiplies  the. Quotient  hy  that  Factor, 

4.  Dividing  the  Dividend  hy  any  Factor  in  effect  di- 
vides the  Quotient  hy  that  Factor, 

5.  Multiplying  the  Divisor  hy  any  Factor  in  effect  di- 
vides the  Quotient  hy  that  Factor, 

6.  Dividing  the  Divisor  hy  any  Factor  in  effect  multi- 
plies  the  Quotient  hy  that  Factor. 

7.  Multiplying  hoth  Dividend  and  Divisor  hy  the  same 
Factor  does  not  affect  the  Quotient, 

8.  Dividing  hoth  Dividend  and  Divisor  hy  the  same 
Factor  does  not  affect  the  Quotient, 


MENTAL  AND    WRITTEN  ARITH3IETIC.  149 

Dividing  both  Dividend  and  Divisor  by  the  same 
Factor  is  the  same  as  rejecting  that  Factor  from  both. 

DEFINITION, 

Cancellation  is  rejecting  equal  Factors  from  both 
Dividend  and  Divisor. 

Example.  Divide  60  by  15. 

SOLUTION. 

Dividend.    60       $x0x4        .^^.^ 

—  =  — —  —  4.  Quotient. 

Divisor.       15  ?  x  ^ 

ExPLAiq-ATiON".  Factoring  the  Divisor  into  3  and  5, 
and  the  Dividend  into  3,  5  and  4,  and  rejecting  the 
Factors  3  and  5  from  both,  by  Principle  8,  we  have  4 
for  the  Quotient. 

By  Cancellation  find  the  Quotients  in  the  following 

Exercises  for  the  Slate  akd  Board. 

1728 
144* 

When  the  Divisor  is  contained  in  the  Dividend  with- 
out a  Eemainder,  the  Divisor  is  named  an  Exact 
Divisor. 

Example.  Divide  30  by  42. 

SOLUTION. 

Dividend.    30       ^  x  $  x  5       5     . 

zzz  — r^:  Jjin^ 

Divisor.       42       ^  x  $  x  7       7 

ExPLANATioiq-.  Factoring  and  canceling,  we  find  no 
Factor  7  in  the  Dividend.     Hence, 

General  Principle  in  Division, 

9.  The  Dividend  cannot  he  exactly  divided  hy  the 
Divisor  wlien  any  Factor  of  the  Divisor  is  not  found  in 
it. 


48 

70 

168 

210 

385 

735 

12' 

14' 

U' 

30' 

35' 

105' 

150 


FIRST  LESSONS  IN 


1 


LESSON   GIL 

Example.  Clifford's  mother  divided  a  watermelon 
between  him  and  his  sister.    What  did  each  receive  ? 

ExPLANATiOiq-.  Writing  the  Dividend 
and  Di^dsor  as  in  former  cases,  we  find  solution. 

that  we  can  not  so  factor  the  Dividend       Dividend,  1 
as  to  obtain  a  Factor  2.    Hence,  accord-        Divisor,     2 
ing  to  Principle  9,  1  can  not  be  exactly 
divided  by  2. 

But  since  there  are  2  children,  and  there  is  only  1 
melon,  it  is  evident  that  the  melon  must  be  divided  into 
2  equal  'parts,  and  1  'part  given  to  each  child. 

When  any  single  thing  is  divided  into  2  equal  parts, 
these  parts  are  named  halves.  One  of  these  is  called 
one  Half,  and  is  written  |. 

If  an  apple  be  cut  into  S  equal  parts,  these  parts  are 
named  Thirds.  One  part  is  named  one  Third,  and 
written  ^,     Two  Thirds  are  written  f . 


MENTAL  AND    WRITTEN  ARITHMETIC.  151 

If  a  pear  be  divided  into  ^  equal  parts,  these  parts  are  ' 
named  Fou7^ths.      One  Fourth  is  written  |;    two 
Fourths  are  written  | ;  and  three  Fourths  J. 

If  5  oranges  are  to  be  divided  between  2  children,  we 
can  give  each  child  2  oranges,  that  is  4  oranges  to  the 
2  children,  and  then  divide  the  fifth  orange  into  2 
Halves,  and  give  1  Half-orange  to  each  child.  Each 
child  would  then  have  2  oranges  and  1  Half-orange ; 
which  are  written  ^|  oranges. 

When  a  watermelon  is  divided  into,  2  equal  parts,  or 
an  apple  into  3  equal  parts,  the  melon  or  apple  is  cut  or 
fractured,  and  one  of  the  parts  is  2i  fragment,  or  Frac- 
tion  of  the  entire  thing.  Hence,  one  Half,  one  Third, 
one  Fourth,  two  Thirds,  three  Fourths,  or  ^,  I,  \,  |,  |, 
are  named  Fractions. 

DEFINITIONS, 

1.  An  Integral  Unit  is  a  single  entire  thing. 

2.  A  Fractional  Unit  is  one  of  the  equal  parts 
into  which  an  Integral  Unit  is  divided. 

S.  An  Integral  Number ^  or  Integer^  is  an 
Integral  Unit  or  collection  of  Integral  Units. 

4.  A  Fractional  l^umber^  or  Fraction^  is  a 
Fractional  Unit,  or  a  collection  of  Fractional  Units. 

5.  A  Mixed  Number  is  a  number  consisting  of 
loth  an  Integer  and  a  Fraction. 

Eemakks. 

1.  Integral  means  ivhole,  or  entire.  Hence,  an  Integer 
is  frequently  called  a  Whole  Number. 

2.  An  Integral  Unit  is  commonly  called  simply  a 
Unit ;  or,  sometimes,  a  If  nit  One. 


152 


FIRST  LESSONS  IN 


LESSON    CIIL 

Suppose  we  wish  to  divide  3  apples  equally  between 
2  persons.  1st :  It  is  evident  that  we  can  divide  each 
apple  into  2  equal  parts,  and  give  each  person  1  part 
from  each  apple.  He  would  have  as  many  parts  as  there 
were  apples ;  or  3  parts.  He  would  receive  3  Halves ; 
or  |.  2d :  If  we  chose,  however,  we  might  divide  the  3 
apples  between  the  2  persons  by  at  first  giving  each  1 
apple,  or  2  apples  to  both,  and  then  cutting  the  third 
apple  into  2  equal  parts  and  giving  1  part  to  eaeh. 
Each  person  would  thus  have  1  entire  apjole,  and  1  Half- 
apple  ;  or  1^  apples.  It  is  plain  that  each  person  would 
receive  the  same  in  both  cases ;  hence,  |  are  the  same  as 
1^.  This  must  be  evident.  For,  since  2  of  the  3  Halves 
will  make  one  apple,  3  Halves  are  the  same  as  1  \, 

If  4  apples  were  to  be  divided  among  3  persons,  we 
might  cut  each  apple  into  3  pieces,  and  give  each  person 
a  piece  from  each  apple.  Each  person  would  then  have 
4  pieces,  or  |.  And,  since  3  Thirds  make  1  apple,  he 
would  have  the  same  as  1  apple  and  one  Third  of  an 
apple;  or>l|  apples. 


MENTAL  AND    WRITTEN  ARITHMETIC.  153 

In  the  expressions,  |  and  |,  the  Dividends,  3  apples 
and  4  apples,  show  the  number  of  things  divided ;  and, 
since  each  person  has  1  piece  from  each  thing  divided, 
^3  and  4  also  show  the  number  of  parts  each  person  re- 
ceives. The  Divisors,  2  and  3,  show  the  name  or  kind 
of  the  parts  received  by  each  person.  Name  means  the 
same  as  Denomination. 

The  Expressions  |  and  |  are  Fractions.    Hence, 

DEFINITIONS. 

6.  The  DiviDEKD  in  a  Fraction  is  named  the  Nu" 
meratoVf  because  it  tells  the  Number  of  parts  in  the 
Fraction, 

7.  The  Divisor  in  a  Fraction  is  named  the  Denom- 
inatOTf  because  it  tells  the  ^N^ame,  or  Dekomikatiok, 
of  the  parts  in  the  Fraction, 

8.  The  Numerator  and  Dekomikator,  taJcen  to- 
gether, are  named  the  Terms  of  the  Fraction, 

9.  The  LINE  {Sign  Minus)  loritten  between  the  Terms 
of  a  Fraction  is  named  the  Dividing -line ^  because 
it  shows  thdt  the  Numerator  is  to  be  divided  by  the 
Denominator, 

10.  The  Value  of  a  Fraction  is  the  Quotient  aris- 
ing from  dividing  the  Numerator  by  the  Denominator, 

11.  A  Minor  Fraction  is  a  Fraction  whose  value 
is  LESS  THAN  the  Unit  One. 

12.  A  Major  Fraction  is  a  Fraction  whose  value 
EQUALS  OR  EXCEEDS  the  Unit  One  ;  that  is,  whose  value 
is  greater  than  that  of  any  Minor  Fraction. 

Eemark. 
Minor  means  less,  or  smaller;   and  Major  means 
greater, 

13.  Like  Fractions  are  Fractions  having  like  or 
equal  Denominators. 


154  FIRST  LESSONS  IN 

14  Tinlike  Fractions  are  Fractions  having  uk« 

liIKE  OR  UifEQUAL  DeKOMIJS^ATORS. 

Kemark. 
f  and  4  are  Like  Fractions,  |  and  /y  Unlike. 
15.  To  Reduce  a  Fraction  is  to  change  its 

FORM  without   CHAKGIi^G   ITS   VALUE. 

Example  1.  How  many  apples  in  ^^-  apples? 

ExPLAKATioiT.  Since  2  Halves  make  one  apple,  we 
shall  have  as  many  apples  as  2  Halves  are  contained 
times  in  12  Halves ;  which  are  6  times.    Hence  ^^  =  6. 

Example  2.  How  many  apples  in  -Y-  apples  ? 

ExPLAN^ATiO]sr.  In  16  Thirds  there  are  as  many  Ones 
as  3  Thirds  are  contained  times  in  16  Thirds.  3  are  in 
16  5  times,  with  1  for  a  Remainder.  Hence,  16  Thirds^ 
or  J36,  are  equal  to  5  Units  and  1  Third;  or  5|.    Hence, 

To  Reduce  a  Major  Fraction  to  an  Integer, 
OR  Mixed  Number: 

Eule. 

Divide  the  Numerator  of  the  Fraction  ly  the  Denomi- 
nator; and  if  there  is  a  Remainder  use  it  for  the  Numer- 
ator of  a  Fraction,  with  the  Divisor  for  Denominator, 
and  annex  this  Fraction  to  the  Quotient. 

Eeduce  the  Fractions  to  Integers  or  Mixed  Numbers 
in  these 

Exercises  for  the  Slate  akd  Board. 
I. 


11 

17    39    148 

739 

90    360 

540 

2' 

5'   8'    6  ' 

7' 
II. 

9'    6' 

90  ■ 

84 

795    1189 

1973 

19000 

7354 

21' 

25  '    32  ' 

176  ' 

1700  ' 

679 

MENTAL  AND    WRITTEN  ARITHMETIC,  155 

LISSON    CIV. 

Example  1.  In  5  apples  how  many  Thirds  ? 

ExPLAi^ATio^sr.  Since  there  are  3  Thirds  in  1  apple, 
in  5  apples  there  are  5  times  3  Thirds ;  which  are  15 
Thirds;  or  J/*  C)r,  since  there  are  3  times  as  many 
Thirds  as  there  are  apples,  we  may  find  the  number  of 
Thirds  by  multiplying  the  number  of  apples  by  3.  3 
times  5  are  15.  Hence,  there  are  15  Thirds;  or  -*/-. 
Therefore, 


To  Reduce  an  Integer  to  the  form  of  a 
Fraction: 

EULE. 

Multiply  the  Integer  dy  the  Denominator  of  the  re- 
quired Fraction,  and  under  this  Product,  used  as  the 
Numerator  of  the  result,  write  the  required  Denominator, 

Exercises  for  the  Slate  and  Board. 
Eeduce  7  to  Thirds ;        13  to  Fifths ;        37  to  Ninths. 
Keduce  123  to  25ths ;       527  to  75fchs ;       317  to  llths. 

Example  2.  In  8|  apples  how  many  Thirds  ? 

Explanatio:n".  Keducing  8,  or  8  apples,  to  Thirds,  by 
the  preceding  Eule,  we  have  (3  times  8  are)  24  Thirds. 
We  have  also  2  other  Thirds  (|).  Adding  24  Thirds 
and  2  Thirds,  we  have  for  a  result  26  Thirds ;  or  ^f. 
Hence, 

To  Reduce  a  Mixed  Number  to  the  form  of  a 
Fraction: 

Eule. 
Multiply  the  Integer  ly  the  Denominator  of  the  FraC' 
tion,  and  to  this  Product  add  the  Numerator,     Under 
this  Sum,  tised  as  a  Numerator,  write  the  Denominator 
of  the  give7i  Fraction  for  a  Denominator. 
11 


156  first  lessons  in 

Exercises  for  the  Slate  akd  Board. 

I. 

5|;      111;      19«3;      23i ;      265,\;      IS?/^;      378^. 

II. 
18311;        72311;        618i||;        372|if;        958?i|. 

If  James  has  5  oranges  and  John  3  oranges,  we  find 
how  many  they  both  have  by  adding  together  5  and  3 
and  obtaining  their  Sum,  8.  So  if  they  have  things  of 
any  other  kind,  and  the  things  which  they  both  have 
are  of  the  same  hindy  we  find  how  many  they  both  have 
by  adding  the  numbers  showing  how  many  each  has. 

Example  3.  Frank  has  |  of  a  watermelon,  and  Harry 
has  f  of  it.    What  have  both  ? 

ExPLAKATioiT.  Since  Frank  had  3  pieces  and  Harry 
2  pieces,  and  both  had  pieces  of  the  same  hind,  or  size, 
we  add  3  pieces  and  2  pieces,  and  have  5  pieces,  or  |. 
Hence, 

To  Add  Like  Fractions: 

Eule. 
Find  tJie  Sum  of  the  Numerators  of  the  Fractions, 
and,  using  this  for  the  Numerator  of  the  result,  write  the 
common  Denominator  for  a  Denominator, 

Exercises  for  the  Slate  akd  Board. 

5.3^,    A  .  1^.    'l5  +  l?^.    11  +  ^  =  ? 
9       9       *     17       17       *     25      25       *     32      32 

II. 

35      35       '56      56       *    85      85      *    125      125 

Since  Subtraction  is  the  reverse  of  Addition,  from  our 
Rule  for  the  Addition  of  Fractions  we  must  have 


MENTAL  AND    WRITTEN  ARITHMETIC. 


157 


To  Subtract  a  Fraction  from  a  Like  Fraction: 

KULE. 

Subtract  the  Numerator  of  the  SuUrahend  from  that 
of  the  Minuend,  and,  using  the  Difference  as  the  Numer- 
ator of  the  result,  write  the  common  Denominator  for  a 
Denominator. 


Exercises  for  the  Slate  and  Board. 


7  4 
---  =  ? 

8  8 


13       13 


39 

27 


18 

27 


==::? 


49 
53 


53 


LESSON   GV. 


Example.  Walter's  mother  gave  him  ^  of  a  cake,  and 
f  of  another  cake  of  the  same  size.  What  had  he  in 
all? 


rmST  CAKE. 


SECOND  CAKE. 


Explanation.  In  the 
cut,  at  the  right,  we  see 
the  first  cake  cut  into 
Halves,  and  the  Half 
given  to  Walter  placed 
below  it. 

We  see  also  the  second 
cake  cut  into  Thirds, 
and  the  ^  Thirds  given 
to  Walter  placed  below. 

We  observe  that  the  1 
Half  and  the  2  Thirds 
are  not  parts  of  the  same 

hind,  or  size,  and  hence  cannot  he  counted  together,  or 
added.    We  must  cut  the  1  Half  into  smaller  parts,  and 


158 


FIRST  LESSONS  IN 


also  the  2  Thirds  into  smaller  parts^  in  such  manner 
that  all  the  parts  shall  be  of  the  same  size. 

Cutting  the  1  Half 
into  3  equal  parts,  as 
shown  at  the  right,  we 
see  that  there  would  be 
6  such  parts  in  the 
whole  cake.  Hence  i 
of  the  cake  is  the  same 
as  §  of  the  cake. 

Cutting  each  of  the  2  Thirds  into  2  equal  parts,  they 
make  4  parts.  There  would  be  6  such  parts  in  the 
second  cake.  Hence  the  4  parts  are  Sixths  ;  and  it  fol- 
lows that  f  of  this  cake  are  the  same  as  %  of  it. 

Therefore  Walter  had  |  of  the  first  cake,  and  %  of  the 
second  cake.  Since  these  parts  are  all  of  the  same  size, 
they  can  be  added  by  the  Eule  in 
the  last  Lesson.  Adding  them 
according  to  the  Eule,  |  and  | 
are  |.  This  is  a  Major  Fraction. 
Keducing  it  to  a  Mixed  ^NTumber 
according  to  the  Eule  on  page 
154,  we  have  1^.  This  is  shown 
in  the  cut  at  the  right,  by  placing 
the  3  Sixths  and  4  Sixths  together. 
6  of  the  7  Sixths  make  a  whole  cake,  and  the  remaining 

1  Sixth  is  placed  on  the  top  of  this  cake.    Hence  Walter 
received  1  cake  and  1  Sixth  of  a  cake. 

Thus  we  have  first  changed  h  into  |,  and  |  into  |, 
and  then  added  them  and  obtained  |,  or  IJ. 

The  Numerator  of  {  shows  that  there  is  1  part  in  the 
Fraction  ;  and  the  Denominator,  2,  shows  that  there  are 

2  such  parts  in  one  calce.     So,  also,  in  any  Fraction,  the 
Numerator  shows  the  number  of  parts  in  the  Fraction, 


^^Whs.o^^ 


MENTAL  AND    WRITTEN  ARITHMETIC,  159 

and  the  Denominator  shows  the  number  of  such  parts  in 
a  Unit,  Hence,  when  we  cut  the  1  part  in  the  Numer- 
ator of  ^  into  3  parts,  there  will  be  S  times  as  many 
such  parts  as  there  are  Halves  in  the  cake ;  or  3 
times  2  parts,  which  are  6  parts.  Hence,  cutting  our  ^ 
cake  into  |,  the  Numerator  and  Denominator  of  |  are 
leach  S  times  as  large  as  the  corresponding  terms  of  i. 
^hat  is,  we  change  |  to  f  hy  multijjlying  loth  its  terms 
ly  S,  This  does  not  change  the  value  of  the  Fraction. 
This  agrees  with  the  7th  Principle  of  Division. 

When  we  cut  each  of  the  2  parts  in  |  into  2  parts, 
changing  the  2  Thirds  to  4  Sixths,  or  |  to  |,  we  make 
both  terms  of  |  tivice  as  large,  or  multiply  both  terms  by 
2,    This  has  not  changed  the  value  of  the  Fraction. 

By  multiplying  both  terms  of  each  Fraction  by  the 
Denominator  of  the  other,  we  have  reduced  the  Unlike 
Fractions  i  and  |  to  the  Like  Fractions  §  and  |. 

In  the  same  manner  we  may  reduce  ^,  |  and  |,  to 
Like  Fractions  by  multiplying  both  terms  of  i  by  the 
Denominators  3  and  4 ;  both  terms  of  |  by  the  Denom- 
inators 2  and  4 ;  and  both  terms  of  |  by  2  and  3. 

If  we  have  any  number  of  Unlike  Fractions,  we  may 
proceed  in  the  same  manner.    Hence, 

To  Reduce  Unlike  to  Like  Fractions: 

EULE. 

Multiply  both  terms  of  each  Fraction  by  each  of  the 
other  Denominators  successively. 

'  Eemark. — It  is  necessary  to  obtain  the  Denominator 
of  only  the  first  reduced  Fraction,  since  all  the  other 
Denominators  are  like  it. 


160 


FIRST  LESSONS  IJST 


Eeduce  Unlike  to  Like  Fractions  in  the  following 

EXEKCISES  FOR  THE   SlATE  AKD   BoARD. 

I. 


I  and  I ; 


and  4; 


3?     5     and    7y    y 

II. 


h  %  I  and  y\  ;  i  f  and  ^3 ; 


3     4  pnrl     8 
3>   lT>  TT  '^'^^  T7« 


LESSON   GVI. 


Example.  Add  5f  and  ^. 

ExPLAi^ATiOK.  Eeducing  | 
and  f  to  Like  Fracticns,  we 
have  ^f  and  ^  ?.  Adding  these, 
we  have  |f,  or  l^J.     Having 


2 

3 

1 4 

2T 


+  ^f 


and  f 

32 
2T? 


or  m. 


5  +  4  +  1=10.     Hence 


of  and  4f 


the  Sum  of  the  Fractions,  we 
add  to  this  the  Integers  5  and  4.  The  Sum  of  5,  4  and 
1  is  10.  Writing  the  Fractional  part  of  the  Sum  after 
this,  we  have  10^  {.    Hence, 

To  Add  Mixed  Numbers: 

EULE. 

Add  the  Fractions,  and  if  their  Sum  is  a  Major  Frac- 
tion reduce  it  to  an  Integer  or  Mixed  Number,  Add  the 
Integral  part  of  this  result  ivith  the  given  Integers,  and 
to  this  Sum  annex  the  Fractional  part. 

Exercises  for  the  Slate  akd  Board. 
I. 

3I  +  3I-?      7|  +  lli=?     12H  +  9J-|=?     Uf  +  2J^T=? 

II- 
4f  +  5i=?       6f  +  8|=?       10J+9|=?       16f  +  13/yrr:? 


MENTAL  AND    WRITTEN  ARITHMETIC.  161 

Example.  From  8|  sub- 
tract 4|.  SOLUTION. 

ExPLANATioif.  Reducing     \=~hy  ^^^  I=t% 
\  and  I  to  the  Like  Frac-      8/5  =  7f§.    Hence 
tions  j%  and  f^,  we  find  that      8|-4|==7f  §-4yV 
y^^    cannot    be    subtracted      ff--y%= j|,  and  7— 4=3. 
from  j\,  since  it  exceeds  it.      Hence  8|  — 4|=3}|. 
Therefore  we  take  one  of 

the  8  Units,  and,  calling  it  ||,  add  it  to  the  ^5,  and  have 
f  f.  Our  Minuend  is  then  7f f,  and  ojir  Subtrahend  4y%. 
Subtracting  y^^  from  ff  we  have  j|  for  the  Fractional 
part  of  our  Eemainder.  Subtracting  4  Ones  from  7 
Ones,  we  have  3  Ones  left.  Uniting  both  parts  of  our 
Remainder,  we  have  3}  |  for  the  true  Remainder.    Hence, 

To  Subtract  a  Mixed  Number,  or  a  Fraction^ 
FROM  A  Mixed  Number  or  an  Integer  : 

Rule. 

I.  Reduce  to  Like  Fractions  the  Fraction  in  the  Sub- 
trahend, and  also  such  part  of  the  Minuend,  {including 
the  Fraction,  if  any,)  as  shall  equal  or  exceed  this. 

II.  Subtract  the  Fractional  part  of  the  Subtrahend 
from  that  in  the  Minuend,  and  the  Integer  in  the  Sub- 
trahend from  that  in  the  Minuend,  and  unite  the  two 
partial  Remainders  into  one  Final  Remainder. 

Exercises  for  the  Slate  and  Board. 

I. 
7|-5|=?       9|-5i=?       ll|-5/y=?       10|-3f=? 

II. 
12J-5|=?      15~64=?      21/7-31=?      35-21/3=? 


162  FIRST  LESSONS  IN 


LESSON   evil. 

MZrZTITZIC^TIOJV*  oiJV^    ^lYISIOJST   01^ 
I^^;ACTIOJVS. 

In  a  Fraction  the  Numerator  is  a  Dividend,  and  the 
Denominator  a  Divisor.  The  Value  of  the  Fraction  is 
the  Quotient.  Hence  we  may  make  any  change,  in  the 
Fraction,  which  does  not  change  the  Quotient.  But, 
according  to  the  8th  Principle  in  Division,  dividing 
both  Dividend  and  Divisor  (Numerator  and  Denomina- 
tor) by  the  same  number  does  not  alter  the  Quotient. 

If  we  take  the  Fraction  /^  and  factor  both  terms,  we 
have  f^.  Dividing  both  terms  by  3,  by  rejecting  the 
common  Factor  3,  we  have  |.  Hence  -f^  equal  |.  The 
terms  of  |  are  smaller  than  those  of  y%,  and  hence  more 
convenient.  We  reduced  y%  to  |  by  rejecting  the  Factors 
common  to  loth  terms, 

DEFINITION. 

A  Fraction  is  expressed  in  its  Lowest  Terms  when 
there  is  no  Factor  common  to  both  terms.    Hence, 

To  Reduce  a  Fraction  to  Lowest  Terms: 

EULE. 

Reject  all  the  CoMMOiq'  Factors  from  both  Terms. 
Exercises  for  the  Slate  akd  Board. 

18.  21.  24.  90.  i3o.  216.  375.         _1  44 

■g"?  ^  2  7  >  35?  Ib^y  f89?  B^O?  600?  ll  ■2'S* 


SOLUTION. 


Example.  Multiply  ^  by  3. 

EXPLAKATIOK.     1st.    Ac-  First  Method. 

cording  to    the    3d    Prin-  i  ^g—i  x  3  —  3—1   ^^^^ 
ciple     in    Division,   multi- 
plying the  Dividend  multi-  Second  MethM. 
plies  the  Quotient.    Hence,  ^x  3  =  ^^-3 =2*  ^^^' 


MENTAL  AND  WRITTEN  ARITHMETIC.  163 

we  multiply  the  Numerator  of  \  by  3,  and  obtain  |  for 
a  Product ;  or,  in  Lowest  Terms,  \. 

2d.  According  to  the  6th  Principle  in  Division,  divid- 
ing the  Divisor  multiplies  the  Quotient.  Hence  we 
multiply  J  by  3  by  dividing  the  Denominator  by  3,  and 
have  \  for  our  Final  Product,  as  before.    Hence, 

To  Multiply  a  Fraction  by  an  Integer: 

EULE. 

Multiply  the  Numerator  of  the  Fraction  ly  the  Integer, 
and  for  a  Denominator  write  the  given  Denominator  ;  or, 
Divide  the  Denominator  of  the  Fraction  by  the  Integer, 
and  write  the  given  Numerator  over  this  for  a  Numer- 
ator* 

Exercises  for  the  Slate  aitd  Board. 

First  Method, 

J  X  3  =  ?      I  X  2  =  ?      /_  X  4  =  ?      3^  X  11  =  ? 

Second  Method. 

§  X  3  =  ?     Vo"?  X  7  =  ?     4^  X  9  =  ?     /g  X  8  =  ? 


BOLTTTION. 

First  Method. 


Example.    Divide  f  by  3. 

Explanation.  According 
to  the  4th  General  Princi-        ^^^^^^^^^  Quotient 
pie  in  Division,  dividing  the       ^  '        ^        ^'  ^ 
Dividend  divides  the  Quotient. 

Hence,  we  divide  the  Numerator  of  f  by  3,  and  obtain 
f  for  our  Quotient. 

According  to  the  5th  General  Principle  in  Division, 
multiplying  the  Divisor  divides  the  Quotient.     Hence, 
multiplying  the  Deno- 
minator of  f  by  3,  we  h  ave  second  Method. 

-^j ;  or,  in  Lowest  Terms,      fi~3=f  X3=i/V=f-  Quotient. 
^,  as  before.    Hence, 


164  FIRST  LESSONS  IN 

To  Divide  a  Fraction  by  an  Integer: 

EULE. 

Divide  the  Numerator  of  the  Fraction  ly  the  Integer^ 
and  under  this  Quotient  tvrite  the  given  Denominator 
for  a  Denominator ;  or,  Multiply  the  Denominator  of 
the  Fraction  hy  the  Integer,  and  over  this  Product  write 
the  given  Numerator  for  a  Numerator, 

Eemakk.  The  result  obtained  by  the  Second  Method 
should  be  reduced  to  Lowest  Terms. 

Exercises  eor  the  Slate  akd  Board. 

First  Method. 
Second  Method. 

7y    —    0    —    r  3^-v-O    —    r         y^g    -T-    <    —    f 


LESSON   CVIII, 


Example  1.  Multiply  14  by  4. 

Explain"  ATioiq"     T  o  solution. 

multiply  14  by  4  is  to  14  x  1=14-^7==  V =2.  Prod, 
take  one-seventh  of  14. 

To  take  one-seventh  of  14  we  must  divide  14  by  7.  The 
result,  written  as  a  Fraction,  is  y. 

Example  2.  Mul-  solution. 

tiplyl4by|.  1=1x3. 

ExPLANATioiq-.        14x3=42. 
We  have  seen  that        42  x  4=42^7=-\^=6.  Product, 
when  our  Multiplier 

is  composed  of  Factors  we  can  obtain  the  Product  by 
multiplying  by  the  several  Factors  in  succession.  Find- 
ing that  our  Multiplier  is  composed  of  two  Factors,  we 


MENTAL  AND    WRITTEN  ARITRMETIC.  165 

obtain  our  Product  by  multiplying  by  its  Factors,  3  and 
4.  14  X  3  gives  42.  And  42  x  4  is  the  same  as  42  -^  7, 
which  is  Y"?  ^^  ^• 

Example  3.  Mul-  solution. 

tiply|by-|.  1  =  1x3 

Explanation".      |x3=f''3  =  '5^ 
We  multiply  |  by  3      ^f  x  1  .=J,^-7=J#-x7=l|.  Prod. 
and  ^,  the  Factors  of 

-|.  We  multiply  |  by  3  by  multiplying  the  Numerator 
by  3.  We  multiply  this  result  by  \  by  dividing  by  7 ; 
which  we  do  by  multiplying  the  Denominator  by  7. 
This  gives  \  |  for  the  Product.  Examining  these  Ex- 
amples and  Solutions,  we  see  that  we  multiply  by  a 
Fraction  by  multiplying  the  Multiplicand  by  the  Nu- 
merator of  the  Multiplier,  and  dividing  the  result  by 
the  Denominator  of  the  Multiplier.    Hence, 

To  Multiply  by  a  Fraction: 

EULE. 

Multiply  the  Multiplicand  hy  the  Numerator  of  the 
Multiplier,  and  divide  this  result  hy  the  Denominator. 

Eemarks. 

1.  Mixed  INTumbers  can  be  multiplied  together  by  the 
above  Rule,  after  reducing  them  to  Major  Fractions. 

2.  The  Product  should  be  reduced  to  Lowest  Terms, 
or  to  an  Integer  or  Mixed  Number. 

Exercises  for  the  Slate  an^d  Board. 


Fractions. 

8xi  =  ? 
f  X  1  =  ? 

12  X  1  =  ?      15  X  1  =  ? 
4  X  1  =  ?      tV  X  1  =  ? 

Mixed  Numbers, 

21  X  4  =  ? 

21  x  If  =  ? 

5|  X  31  =  V    71  X  81  =  ? 

9/t  X  3?  =  ? 

166  FIRST  LESSONS  IN 


LESSON   CIX. 

Example  1.  Divide  5  by  i. 

EXPLAIN^ATION.  It  is  plain  solution. 

that  4  is  contained  in  1  seven  5-h^=5  x  7==35.  QuoH, 
times.    It  must  be  contained 

in  2  twice  7  times,  and  in  5  five  times  7  times,  or,  which 
is  the  same,  7  times  5  times;  which  are  35  times. 
Hence  we  divide  5  by  ^  by  multiplying  5  by  7. 

Example  2.    Divide  5  by  |. 

Explanation.  We  solution. 

factor   our  Divisor   |  1  =  ^x3 

into  1  and  3.    Divid-  5-^3=| 

ing  first  by  the  Factor        |-^|=|  x  7=|^^  =  V"-  Q^oH. 

3,  we  have  |.     Divid- 
ing this  result  by  the  other  Factor,  4,  by  multiplying 
by  7,  we  have  y  for  the  result,  or  Quotient. 

Example  3.  Divide  |  by  f . 

Explanation.  Factor-  solution. 

toring  f  into  4  and  5,  we  |=:X  x  5 

divide  |  by  5,  by  multi-  |4-5=:|x5=^^o 

plying   the   Denominator        •/o"^t=2U  x '^=^%^^=i0 
by  5,  and  have  for  a  result        f  i^lsV*  ^''^^' 
5%.    We  then  divide  ^%  by 

4,  by  multipljring  by  7.  This  gives  f  J  for  the  Quotient. 
Eeducing  f  J  to  a  Mixed  Number,  we  have  1-^^  for  our 
final  Quotient. 

Examining  these  Solutions  and  Explanations,  we  see 
that  we  have  in  each  case  divided  by  a  Fraction  by 
dividing  the  Dividend  by  the  Numerator  of  the  Divisor, 
and  then  multiplying  this  result  by  the  Denominator 
of  the  Divisor.    Hence, 


MENTAL  AND  WRITTEN  ARITHMETIC.  167 

To  Divide  by  a  Fraction: 

Rule. 

Divide  the  Dividend  hy  the  Numerator  of  the  Divisor^ 
and  then  multiply  this  result  hy  the  Denominator  of  the 
Divisor, 

Eemakks. 

1.  The  result  should  be  reduced  to  Lowest  Terms ;  if 
a  Major  Fraction,  to  an  Integer  or  Mixed  Number. 

2.  Mixed  Numbers  in  the  Dividend  or  Divisor  may 
be  reduced  to  Major  Fractions,  and  then  the  Division 
be  performed  by  the  above  Eule. 

Exercises  for  the  Slate  akd  Board. 

Fractions, 

9~|  =  ?      8-f-|  =  ?      15-^f=?      l-r-|=? 

4      •      3    —    *  g      •      ^    —    •  IT      •      4    —    ^  25      •     Id   —    • 

Mixed  Numbers, 

2|-  -^-  3|  =  ?    5|  -V-  7|  =  ?    12|  ~  4^  =  ?    3|  -^  If  =  ? 


To  Teachers. — The  Elementary  Principles  involved 
in  the  Reduction  of  Denominate  Numbers,  the  Addition 
and  Subtraction  of  Compound  Numbers,  and  the  Mul- 
tiplication and  Division  of  Compound  by  Simple  Num- 
bers, are  the  same  which  are  used  in  Addition,  Subtrac- 
tion, Multiplication  and  Division  of  Simple  Numbers, 
and  have  already  been  fully  set  forth.  In  applying  them 
to  Denominate  and  Compound  Numbers,  the  slight 
changes  necessary  to  be  made  can  be  readily  and  easily 
explained  by  Teachers.  Therefore,  the  Examples  here- 
inafter given  are  not  accompanied  with  explanations. 


168  FIRST  LESSONS  IN 


DENOMINATE  NUMBERS. 


LESSON   OX. 

In  measuring  a  quantity,  we  take  some  definite  amount 
of  it  for  a  Unit ;  as  a  pint,  a  yard.  We  often  have  dif- 
ferent Units  for  the  same  kind  of  quantity. 

I^EFINITIONS. 

1.  A  TTnit  of  Measure  is  the  definite  amount  of 
anything  taken  as  a  stakdaed  of  comparison  in  meas- 
uring all  quantities  of  that  hind. 

2.  Denoinifiation  is  the  name  given  to  a  Unit  of 
Measure  ;  as  quart,  ounce,  shilling, 

3.  A  Higher  Denoi^nination  is  that  one  of  two 
Denominations  whose  Unit  has  the  higher  yalue. 

4.  A  Lower  Denomination  is  that  one  of  two 
Denominations  whose  Unit  has  the  lower  yalue. 

5.  A  Denominate  JS'umber  is  a  number  applied 
to  one  or  more  Denominations  ;  as  4-  gallons,  3  quarts, 

6.  A  Simple  Number  is  a  number  expressed  either 
in  NO  Denomination  or  only  one. 

7.  A  Compound  NiiTnher  is  a  number  expressed 
in  more  than  one  Denomination  ;  as  2  days  5  hours, 

8.  deduction  of  Denominate  Numhe^^s  is 
changing  the  number  and  Denomination  of  a  Denom- 
inate Number  without  changing  its  Value. 

9.  Reduction  Ascending  is  reducing  a  Denom- 
inate Number  to  one  of  a  Higher  Denomination. 

10.  Meduction  Descending  is  reducing  a  De- 
nominate Number  to  one  of  a  Lower  Denomination. 


MENTAL  AND    WRITTEN  ARITHMETIC.  169 


LESSON   CXI. 

UJSriTBD    ST  A.  TBS  MOJSTBT. 

Tfnited  States  Money ^  called  also  Federal 

money  f  is  the  legal  currency  of  the  United  States. 

TABZE. 

10  mills  {m),  are  1  cent.  d. 

10  cents  are  1  dime.  d. 

10  dimes,  or  100  ct.,  are  1  dollar.  $. 

10  dollars  are  1  eagle.  E, 

Note. — The  Table  of  Canada  Money  is  the  same  as  that  of 
United  States  Money. 

Exercises  in  Reduction, 

In  9  ct.  how  many  mills  ? 

In  8  ct.  7  m.  how  many  mills  ? 

In  7  d.  how  many  cents  ?     How  many  mills  ? 

In  7  d.  8  ct.  9  m.  how  many  mills  ? 

How  many  cents  in  80  m.  ?    In  50  m.  ?    In  90  m.  ? 

How  many  cents  and  mills  in  98  m.  ?    In  65  m.  ? 


170  FIRST  LESSONS  IN 

Addition, 

What  is  the  Sum  of  4  ct.  5  m.,  and  3  ct.  2  m.? 
What  is  the  Sum  of  6  ct.  8  m.,  and  5  ct.  7  m.? 
What  is  the  Sum  of  8  d.  9  ct.  7  m.,  and  3  d.  8  ct.  6  m.  i 

Subtraction, 

From  8  ct.  9  m.  subtract  5  ct.  3  m.? 
Prom  7  d.  8  ct  9  m.  subtract  3d.  4  ct.  5  m. 
From  5  d.  4  ct.  7  m.  subtract  2  d.  9  ct.  3  m. 
From  8  d.  3  ct.  4  m.  subtract  3  d.  5  ct.  7  m. 


LESSON   CXII. 

JEJSTGZISir  MOJSTBT. 


Bnglish  Money ^  called  also  Sterling  Money , 

is  the  currency  of  Great  Britain. 

TABJjE. 

4  farthings  {far,)  are  1  penny.  d. 

12  pence  are  1  shilling.  s, 

20  shillings  are  1  pound  or  sovereign.  £, 

21  shillings  are  1  guinea.  guin. 

Exercises  in  Reduction. 

How  many  farthings  in  7d.  ?     In  9s.  ?    In  3s.  9d.  ? 

How  many  pence  in  7s.  ?     In  £3  ?     In  £6  7s.  9d.  ? 

How  many  farthings  in  £7  lis.  5d.  ?  In  £9  7s.  8d. 
3  far.  ? 

How  many  shillings  in  48d.  ?  In  96  far.  ?  In  240 
far.? 

Keduce  987  far.  to  Higher  Denominations. 

Eeduce  lis.  and  1,765  far.  to  Higher  Denominations. 

In  21  pounds  how  many  guineas  ? 


MENTAL  AND    WRITTEN  ARITHMETIC.  171 

fFTi         -^^^y?^^ — ■ — p 


LESSON   CXI/f. 

liiqttid  Measure^  called  also  Wine  Measure^ 

is  used  in  measuring  wines,  oil^  molasses,  milk,  and 
other  liquids. 


TABLE, 


4    gills  igi.) 

2    pints 

4    quarts 
31^  gallons 

2    barrels,  or ) 
63    gallons,     ) 


are  1  pint, 
are  1  quart, 
are  1  gallon, 
are  1  barrel. 


qt. 
gal 


are  1  hogshead.    JiM. 


Exercises  for  the  Slate  asd  Board. 

Meduction, 

How  many  gills  in  5  pt.  ?    In  7  qt.  ?    In  15  gal.  ? 
How  many  gills  in  15  gal.  3  qt.  1  pt.  2  gi.  ? 
Eeduce  547  gi.  to  Higher  Denominations- 

Addition^ 

To  19  gal.  3  qt.  1  pi  3  gi.  add  9  gal.  2  qt.  1  pt.  2  gi.  ? 
To  20  gal.  2  qt  1  pt.  2  gi.  add  10  gaL  3  qt.  1  pt  3  gL  ? 
12 


173 


FTRST  LESSONS  IN 


Subtraction, 

From  27  gal".  3  qt.  1  pt.  3  gi.  take  19  gal.  2  qt.  1  pt.  2  gi. 
From  9  gal.  0  qt.  1  pt.  1  gi.  take  2  gal.  2  qt.  0  pt.  3  gi. 

Multiplication , 

How  mucli  oil  is  there  in  5  casks,  each  containing  12 
gal.  2qt.  1  pt.  3gi.? 

In  3  casks,  each  containing  10  gaL  1  qt.  1  pt.? 

Division,. 

If  18  gal.  3  qt.  1  pt.  2  gL  of  milk  be  equally  divided 
between  two  persons,  what  will  each  receive  ? 


LESSOIV   CX/V. 


Dry  Measure  is  used  in  measuring  grain,  fruits, 
roots,,  seeds,  salt,  lime,  charcoal,  and  various  other  ar- 
ticles not  fluid. 

TABZE. 

2  pints  (pt.)  are  1  quart.  qf. 

8  quarts  are  1  peck.  ph 

4  pecks  are  1  bushel.  hu. 

36  bushels  (of  coal)  are  1  chaldron*  chal. 


MENTAL  AND    WRITTEN  ARITHMETIC.  173 

EXEECISES  FOR  THE   SlATE  AKD   BoAKD. 
A  ddition. 

Add  5  bu.  3  pk.  7  qt.  1  pt.  and    3  bu.  2  pk.  4  qt.  1  pt. 
Add  7  bu.  1  pk.  5  qt.  1  pt.  and  11  bu.  3  pk.  7  qt.  1  pt. 

Subtraction, 

From  31  bu.  3  pk.  7  qt.  0  pt.  take  27  bu.  3  pk.  4  qt.  1  pt. 
From  25  bu.  3  pk.  0  qt.  0  pt.  take  16  bu.  2  pk.  5  qt.  1  pt. 


LESSON    CXV. 

Avoirdupois  Weight  is  used  for  ail  the  ordinary 
purposes  of  weighing, 

TABLE, 

are  1     ounce.         oz. 

are  1     pound.        Ih 

are  1     quarter.      qr. 

^  ( hundred-  ] 
are  1  i  _^,.^i.^^     j- 


16  drams  {dr,) 
16  ounces 
25  pounds 
4  quarters,  or  \ 


100  pounds, 


I  weight. 


cwU 


20  hundred  weight  are  1     ton. 


T, 


174  FIRST  LESSONS  IN 

EXEBCISES  FOR  THE   SlATE  A.^B  BoARD. 
JReduction* 

How  many  drams  in  3  qr.  18  lb.  13  oz.  11  dr.  ? 
Eeduce  1572  dr.  to  Higher  Denominations. 

jLddition, 

Tolqr.  171b.    9  oz.    7  dr.  add  1  qr.  15  lb.  13  oz.  14  dr. 
To  3  qr.  21  lb.  11  oz.  13  dr.  add  2  qr.    9  lb.    8  oz.    7  dr. 


LESSON   GXVI. 


Troy  Weight  is  used  in  weighing  jewels,  gold  and 
silyer. 

TJLBZB. 

24  grains  (gr,)      are  1  pennyweight,  ptvt. 
20  pennyweights  are  1  ounce.  oz. 

12  ounces  are  1  pound.  lb. 

Note.  The  Treacher  will  supply  under  this  and  the  following 
Tables  all  needed  Exercises. 

Apothecaries^  Weight  is  used  by  physicians  in 
compounding  medicines ;  but  when  medicines  are 
bought  or  sold  Avoirdupois  Weight  is  used. 

TABLE. 

20  grains  {gr.)  are  1  scruple,  sc.  or  3. 

3  scruples       ^re  1  dram.  dr.  or  3  . 

8  drams           are  1  ounce.  oz.  or  § . 

12  ounces          are  1  pound,  lb.  or  ft. 


MENTAL  AND    WRITTEN  ABITMMETIC. 


17b 


LESSON      CXVII. 

Linear  Measure — called  also  Long  Measure 

— is  used  in  measuring  lines,  or  distances. 


12    inches  (m.) 

3    feet 

51  yards,  or  1Q\  it, 
40    rods 


TAJBZE, 

are  1  foot  ft 

are  1  yard.  yd. 

are  1  rod,  perch,  or  pole.  rd. 

are  1  furlong.  fur. 


8    furlongs,  or  320  rd.,are  1  mile.  mi. 

SQUA^RB   MBASU^B. 

Square  Measure  is  used  for  measuring  surfaces ; 
as  of  land,  plastering,  and  paving. 

A  square  foot  is  a  square  each  of  whose  4  sides  is  1 
foot,  or  12  inches,  in  length.  A  square  yard  is  a  square 
each  of  whose  4  sides  is  1  yard,  or  3  feet,  in  length. 


176 


FIBST  LESSONS  IN 


TABLE, 

144    square  inches  {sq.  in,)  are  1  square  foot.  sq,ft 

9    square  feet  are  1  square  yard.  sq.  yd. 

30|  square  yards  are  1  square  rod.  sq,  rd, 

40    square  rods  are  1  rood.  R, 

i   roods  or)  avelaove.  A. 

160    sq.  rods,  ) 

640    acres  are  1  square  mile,  sq,  mi. 


LESSON   CXVIII. 


Cubic  Wen  sure  is  used  for  measuring  solids;  as 
timber,  wood,  and  stone. 

A  cubic  foot  is  a  cube  each  of  whose  12  edges  is  1 
foot,  or  12  inches,  in  length.  A  cubic  yard  is  a  cube 
measuring  1  yard,  or  3  feet,  on  each  edge.  Each  of  its 
6  faces  is  a  square  containing  9  square  feet.  A  cubic 
yard  is  shown  in  the  above  cut. 


MENTAL  AND    WRITTEN  ARITHMETIC 


177 


TABLE. 

1728   cubic  inches  are  1  cubic  foot.  cu,p, 

27    cubic  feet      are  1  cubic  yard.  cu,  yd, 

42    cubic  feet      are  1  ton,  shipping.  t  s. 

24:|  cubic  feet      are  1  perch,  of  ^tone.  pch. 


Wood  Measure^  though  part  of  Cubic  Measure, 
is  sometimes  embraced  in  a  separate  Table. 

A  pile  of  wood  -8  ft  long,  4  ft.  wide,  and  4  ft.  high, 
contains  a  cord. 

TABLE, 

16  cubic  feet  are  1  cord  foot    cd.fL 

8  cord  feet,  or)      ^re  1  cord.  cd. 

128  cubic  feet,     ) 


EXEECISES  FOR  tHE  SlATE  AND  BOARD. 

In  1  cd.  5  cd.  ft.  11  cu.  ft.  187  cu.  in.,  how  many 
cubic  inches  ? 

Reduce  3  cu.  yd.  5  cu.  ft.  127  cu.  in.  to  cubic  inches. 
To  3  cd.  7  cd.  ft.  11  cu.  ft.  add  5  cd.  9  cd.  ft.  8  cu.  ft. 


178 


FIRST  LESSONS  IN 


LESSON    CX/X. 

TIM:^  MBASTJ'RB. 

Time  is  Duration  having  a  beginning  and  an  end 
Being  definite,  it  can  therefore  be  measured. 

The  Day  and  Year  are  the  Natural  Divisions  of  Tirne^ 
since  they  are  founded  in  Nature. 

TABLB. 


GO  seconds  {sec) 

are  1  minute. 

min. 

60  minutes 

are  1  hour. 

h. 

24  hoRirs 

are  1  day. 

da. 

7  days 

are  1  week. 

wk. 

365  days,  or) 
52  wk.  1  d.,) 

are  1  common  year. 

yr. 

366  days,  or  > 
52  wk.  2  d.,f 

are  1  leap  year. 

leap  1 

12  calendar  months 

;  are  1  year. 

yr. 

100  years 

are  1  century. 

C. 

Every  fourth  year  in  a  century  (except  sometimes  the 
last)  is  a  leap  year;  as  1804,  1808, 1812. 


3IENTAL  AND    WRITTEN  ARITH3IETIC, 


179 


^ly^isiojsr  OjF  tub  tbah. 


Months.                                           Days. 

Seasons. 

January, 

Jan., 

1st  month,  has  31. 

^  1  Winter. 

February, 

Feb., 

2d  month,  has  28  or  2^ 

March, 

Mar., 

3d  month,  has  31. 

April, 

Apr., 

4th  month,  has  30. 

>■  Spring. 

May, 

May, 

5th  month,  has  31. 

June, 

June, 

6th  month,  has  30. 

] 

July, 

July, 

7th  month,  has  31. 

\  Summer. 

August, 

Aug. 

8th  month,  has  31. 

J 

September,  Sept., 

9th  month,  has  30. 

1 

October, 

Oct., 

10th  month,  has  31. 

>■  Autumn. 

November, 

Nov., 

11th  month,  has  30. 

December, 

Dec, 

12th  month,  has  31. 

Winter. 

Note. — February  has  29  days  in  none  but  leap  years. 

LESSON    CXX, 

Circular  and  Angular  Measure  is  used  in 
measuring  angles,  and  in  the  comparison  of  portions  of 
the  circumferences  of  circles. 


60  seconds  ('') 
60  minutes 
30  degrees 

90  degrees 

360  degrees,  or  ^ 

12  signs,      or  \ 

4  quadrants,  J 


TABTjE, 

are  1  minute, 
are  1  degree. 
1  sign. 

are 


are 


j  1  quadrant,  or  )    quad. 
\  right  angle.       f  r.  a, 

j  1  circi 
I  of  a  ci 


cumference  ) 
circle.  j  ' 


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